SGME RESULTS 02?»! CONVERGENCE RATES AND
ASYMPTOTEB BEHAVMHR FOR. THE LAWS 13F
LARGE NUMBERS

Thesis for the Degree of Ph. D.
MECHIGAN STATE UMVERSITY
WAY KUMAR' ROHATGi
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This is to certifg that the

thesis entitled

Some Results on Convergence Rates and Asymptotic

Behaviour for the Laws of Large Numbers
presented by

Vijay Kumar Rohatgi

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Statistics

(SJ/Lt Jae

Major professor

 

Date August 30, 1967

 

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0-169,

ABSTRACT

SOME RESULTS ON CONVERGENCE RATES AND ASYMPTOTIC
BEHAVIOUR FOR THE LAWS OF LARGE NUMBERS

by Vijay Kumar Rohatgi

Let {Xn: n 2 1} be a sequence of real valued random variables

defined on a fixed but otherwise arbitrary probability Space (0,3,P)

n

and let S 2 X . Let {A } be a sequence of real numbers and [B l
n k=1 k n n

be a non-decreasing sequence of positive real numbers. This thesis

contains a unified approach to the study of rates of convergence of

the sequences [Pr(|S - A I > B 6)} {Pr(sup B-IIS - I > e)} and
n n n ’ k2n k k Ak

[Pr( max ISk - Ak‘ > Bnc)}. We concern ourselves mainly with the

lSkSn

following two cases: (i) when the random variables X i = 1,2,...

1,
are independent and identically distributed with a common law iKX)
and (11) when X1, 1 = 1,2,... are independent and uniformly bounded
by a random variable X (in a certain sense). In either case we
choose Bn = nllr, o < r < 2 and An = 0 or ESn according as

o < r < 1 or 1 S r < 2 and the expectation EX exists and is finitéfr”,
The rate of convergence is expressed either in terms of a convergent
series involving such probabilities or in terms of a convergent

sequence involving such probabilities and a non-negative power of n.

We also study the asymptotic behaviour of the large deviation probability
Pr(|Sn| > Bn) for monotone sequences [Bn}’ Bn * Q, of positive real

1

numbers such that 3; Sn 2 o in the Special case when the X are

I
1 8

independent and belong to the domain of attraction of the normal law.

SOME RESULTS ON CONVERGENCE RATES AND ASYMPTOTIC

BEHAVIOUR FOR THE LAWS OF LARGE NUMBERS

By

Vijay Kumar Rohatgi

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Statistics and Probability

1967

ACKNOWLEDGEMENTS

I wish to express my deep gratitutde to Dr. C. C. Heyde of the
Department of Probability and Statistics, University of Sheffield for
suggesting the problem and for his understanding supervision through-
out the course of this investigation. His continual encouragement and
unreserved assistance, and his comments and suggestions were of immense
help. The material in Chapter IV and that of our paper "A pair of
complementary theorems on convergence rates in the law of large numbers,"
Proc. Camb. Phil. Soc. g; (1967), 73-82, which forms a basic part of
Chapter I, was obtained jointly.

The work on this dissertation was carried out while I was visit-
ing the University of Sheffield from September 1965 to June 1967. I
wish to thank the Department of Statistics and Probability, Michigan
State University for making this visit possible and Professor Gani, for
providing an excellent atmosphere at his Department of Probability and
Statistics, University of Sheffield. The financial support for this
investigation was provided by the U.S. Office of Naval Research under
contract No. Nonr-2587(05) through Michigan State University and I
gratefully acknowledge this support.

Finally, I wish to express my appreciation to my advisor, Pro-

fessor Esther Seiden for her continual encouragement and assistance.

ii

TABLE OF CONTENTS

INTRODUCTION 0 O O O O I O O C O O O O O O O O O O O O O O C I O 1
CHAPTER
I. CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS FOR SUMS OF

INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES . . 4

1. 1 IntrOduction O o o o o o o o o o o o o o o o o o o 4

1.2 Preliminaries. . . . . . . . . . . . . . . . . . . 5
1.3 Main Results . . . . . . . . . . . . . . . . . . . 10
1.4 Proofs . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Examples . . . . . . . . . . . . . . . . . . . . . 27
1.6 Some Results on One-sided Convergence Rates. . . . 28
1.7 Miscellany . . . . . . . . . . . . . . . . . . . . 35
II. CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS FOR INDEPENDENT
RANDOM VARIABLES . . . . . . . . . . . . . . . . . . . . . . 40
2.1 Introduction . . . . . . . . . . . . . . . . . . . 40
2.2 Some Preliminary Results . . . . . . . . . . . . . 41
2.3 Main Results . . . . . . . . . . . . . . . . . . . 47
2.4 Proofs . . . . . . . . . . . . . . . . . . . . . . 49
2.5 Exponential Convergence Rates. . . . . . . . . . . 52
2.6 Miscellany . . . . . . . . . . . . . . . . . . . . 58
III. CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS FOR WEIGHTED
SUMS OF INDEPENDENT RANDOM VARIABLES . . . . . . . . . . . . 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . 63
3.2 Preliminaries. . . . . . . . . . . . . . . . . . . 63

3.3 Main Results . . . . . . . . . . . . . . . . . . . 65

iii

CHAPTER
3.4

3.5

3.6

PrOOfs. O O O O O O O O O O O 0

Some Remarks on the Relationships Between

Results and Results Obtained in Chapter I

Miscellany. . . . . . . . . . .

IV. A LARGE DEVIATION PROBLEM . . . . . . . .

4.1
4.2
4.3

BIBLIOGRAPHY.

Introduction. . . . . . . . . .

Resu1t8 O O O O O O O O O O O I

Results on Iterated Logarithm Type-Behaviour.

iv

PAGE

67

73

74

78

78

81

89

91

INTRODUCTION

Let th: n 2 1} be a sequence of real valued random variables
defined on a fixed but otherwise arbitrary probability Space (0,3,P)

n
and let S a Z Xk. Let {A } be a sequence of real numbers and
n k=1 n

{Bu} be a non-decreasing sequence of positive real numbers. Many

limit theorems of probability theory may then be formulated as theorems

concerning the convergence of one of the sequences [Pr(ISn - Anl > Bne)l,

- > - >
{Pr(sup Bk Sk Akl 3)} and {Pr( max ISk AR] Bne)}, for arbitrary
kzn 1SkSn

e > 0, to an appropriate limiting value. This dissertation contains a
unified approach to the study of rates of convergence of such sequences.
We express the rate of convergence either in terms of a convergent series
involving such probabilities or in terms of a convergent sequence involv-
ing such probabilities and a non-negative power of n. We also study

the asymptotic behaviour of the large deviation probability Pr(ISn| > Bn)
for monotone sequences {Bn}’ Bn d'fl, of positive real numbers such

1

Sn 3 O in the special case when the Xi's are independent

and identically distributed and belong to the domain of attraction of

that B'
n

the normal law.
This dissertation is divided into four Chapters. In the first

Chapter we restrict attention to sequences of independent and identically

distributed random variables xi, 1 = 1,2,... . By analogy with the
KolmogoroveMarcinkiewicz law of large numbers [23, 243] we choose the
l/r

normalizing constants Bn = n , 0 < r < 2, and the centering constants
An = O or ESn according as 0 < r < l or 1 S r < 2 and the expec-

tation exists and is finite. We obtain necessary and sufficient con-

 

 

_
A

 

ditions, in terms of the order of magnitude of Pr(|XI > nl/r), for the
sequences [Pr(|Sn - E Snl > nI/re): n 2 l},
{Pr(sup k’1/r|sk - E skl > e): n 2 1} and {Pr( max Isk - E skl > nl/re): n 2 1}

an 1SkSn

to converge to zero at Specified rates. As a by product we obtain nec-
essary and sufficient conditions for the convergence of the sequence
{Pr(Sn S x)}, for arbitrary x, -O < x.< 9, to zero at specified rates.
‘We also state some results on the rate of convergence of {Pr(Sn S nllrx)}
to zero for -9‘< X‘< 0 and O < r < 2. Some miscellaneous related re-
sults are also given.

In Chapter II we relax the condition of identical distributions
on the sequence {Kn} and assume only that the random variables are in-
dependent. The results obtained here are in the Spirit of those obtained
in Chapter I. More precisely, in the case when the random variables
Xn are uniformly bounded by a random variable X in the sense that
Pr(|Xh| > x) S Pr(‘XI > x) for all x > O, we obtain necessary and
(or) sufficient conditions for the sequences {Pr(|Sn - E Snl > nI/re)},

{Pr(sup k'l/rlsk - E skl > 5)} and (Pr( max Is 1he”

- E skl > n
k2n tsksn

k
to converge to zero at Specified rates. We also generalize the concept
of "exponential convergence" due to Baum, Katz and Read [3] and obtain
necessary and sufficient conditions for exponential convergence of se-
quences of independent random variables. The Chapter concludes with a
section on some miscellaneous related results. The results of this
Chapter extend the previous work [2], [3] on this problem.

In Chapter III we enlarge our sc0pe yet further to consider
weighted sums of independent random variables. Here we investigate the

a
convergence of the sequence {Pr(| 2 an kxkl > 8)} to zero where
k=l ’

3

{an,k: n,k 2 l} is a double sequence of real numbers satisfying certain
mild restrictions. These results complement the work of Franck and
Hanson [10].

In Chapter IV we restrict ourselves to sequences {Kn} of in-
dependent and identically distributed random variables with law iKX).
Let {xn} be a monotone sequence of positive real numbers with xn * a
as n v a such that x;1 Sn ‘ O in probability. The problem of find-
ing the precise asymptotic behaviour of the probabilities Pr(|Sn| > xn)
is a difficult one and it is not possible to solve this problem in as
general a setting as that of Chapter I. However, if X belongs to the
domain of attraction of a stable law it is often possible to give the

precise answer. For example, Heyde [17] showed that if X belongs to

the domain of attraction of a non-normal stable law then
Pr S > x ~ n Pr X > x as n “*w.
(I HI ,9 (I I n),

In this Chapter we consider the case when X belongs to the domain of
normal attraction and shcw that the same asymptotic behaviour is obtained

provided the growth of X is controlled in following manner:
-p
Pr(|X| > x) = x L(x) as x r a,

where p 2 2 and L(') is a function of slow variation, Unfortunately,

a mild growth restriction is also needed on {rm}.

CHAPTER I

CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS FOR SUMS OF

INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES

Section 1.1 Introduction.
Let [an n 2 1} be a sequence of independent and identically
distributed random variables with common law iKX) and write

n

Sn = ZIXk. Take 0 < r < 2. The Kolmogorostarcinkiewicz strong law
k=l'

of large numbers [23, 2437 has the following statement:

l/r

g Elxilr < a, than m“ (3H - n alga—'3' o with

 

,0 lg r<l
c=~‘LE
r x 3; r21
Conversely, if. n.1/r(Sn - n Cr)aA§' 0, then EIXIr < a. ("a.s."

denotes almost sure convergence.)

The purpose of this Chapter is to study the rates of convergence
1 -1
/re)}, [Pr(sup k ”ISk - k C I > 5)}
r
k2n

- k Crl > nl/re)} to zero for arbitrary e > O. The

of the sequences [Pr(|Sn - n Crl > n

and {Pr( max ISk
lsksn

present work extends and unifies the previous work on this problem; the
techniques used, broadly Speaking, date back to Erdbs [8].

In section 1.2 we diSpense with some preliminaries. In section
1.3 we state our main theorems on "two-sided" convergence rates which
we prove in section 1.4. In section 1.5 we give some examples to show
that the two theorems of section 1.3 are complementary in the sense that
neither will consistently give sharper results for a specified generating

type random variables. Section 1.6 is devoted to the study of "one-

Sided" convergence rates. Finally in section 1.7 we present some mis—

A

cellaneous related results.

Section 1.2 Preliminaries.
A function L(x), defined for all sufficiently large x, is said

to be a function of slow variation if, for every c > O as x w w

(1.1) 5331-. 1.

L(x)

The following lemma is due to Karamata [18] (see also Feller
[9, 274]) and is stated here for convenience since we will have occasion

to use it quite frequently in the sequel.

Lemma 1.1. (Karamata)

A_non-negative function, L, of slow variation possesses a_rep-

 

_;esentation.g£ the form

a x x aguz
(1.2) L(x) ='-§—l expgfl u du},

where

a(x) * 1 23_ x r '.

Hereafter L(-) will denote a non-negative, non-decreasing and
continuous function of slow variation. Write 'F(x) = Pr(X S x). The
order Symbols "0" and "0" are frequently used in this and later
Chapters. For an explanation of these symbols we refer to [24, 207]-

We have the

Lemma 1.2.
_If nt+1L(n)Pr(IXI > nl/r) * 0 .ES n v ', then for

a>0,0<vSl,

_ 0(1) if < r(t+1)
(1.3>I _ V,rI2<|°’cIIv‘(x) = OtnnthnV)J'1), n > o ,3; = r<t+1>
|x|<n 0(nvta/r-t-nmnv)]-1) E a > rm).

Proof: Let q(x) = Pr(|X| > x). Then, on integration by parts,

we have

I | I“3F( > = f“v/r “a < >
IxI<nvlrx x -0 x qx

v/r a-l
S a I: x q(x)dx.

Since nt+1L(n)Pr(|Xl > nllr) d O as n w ', given an arbitrarily small
6 > 0, we can choose a constant A1 so large that L(xr)xr(t+1)q(x) < 5
for all x 2 A1. Then, if a < r(t+l), we clearly have

v/r

I: xg-IQ(x)dx = 0(1), while if a 2 r(t+l), we have for fixed

A 2 max(A1,A2) (A to be specified later)

2
v/r‘ v/r
a I: xa-1q(x)dx = a ngg-1q(x)dx +-a I: xg-1q(x)dx

v/r G-l

= 0(1) + or f: x WW)

and
nV/r a-l nv/r a-l-r(t+l) r -1
IA x q(x)dx < 6 I; x _[L(x )] dx
< 6 v r-1an/rxVa/r-l-v(t+l)[L(xv)J-1dx
A
= 6 v r-1[M(n)]-IInv/rxva/r-1-V(t+1)M(n)[M(x)]-1dx

A

where M(x) = L(xv). Since M(x) is also a non-negative function of

slow variation it follows from Lemma 1.1 that for x S n, we have

Mal - 19112.35 “PU: 9%). du},

M(x) - a(x) n

where a(x) d 1 as x r '. Given n > 0 we can choose a constant A

2

sufficiently large that for x 2 A2,

%< (1 + T1) i- exp[(l + Inf: 51511.2}
_ all
— (1 + fl)(x) .
Then, for A 2 max(A1,A2),

Inr/vxva/r-1-\J(t+1)M(n)[M(x)]-1dx
A

r/vao//r-l-v(t-+l)-Tldx

< (1 + mm“ In
A

a > r(t+1)

oz = r(t+1),

and the required result follows.

Lemma 1.3.
m'l/rsn 3 o _i_f_ and only 1;

(i) n Pr(IXI > n ) d 0 2E. n 4 9, and

(ii) nl'I/ff

xdF(x)-'o _a_s_ n-°°°.
le<n1/r .

("P" denotes convergence in probability.)

Proof: Using the degenerate convergence criterion [23, 317],

this result will be established if we can show that (i) and (ii) imply

nl-Z/r lrxzdF(x) w o as n v 9.

|x|<n1

This follows readily from Lemma 1.2.

For the sake of completeness we now state a Simple lemma on in-

finite series which is a straightforward generalization of a well known

8

result (see, for example, [20, 124]).

Lemma 1.4.

Let {an} '22 a non-negative, non-increasing sequence 2f real

numbers converging £2 zero. Let t .25 any real number 2 -1.
a

.EE 2 ntL(n)an < ' then nt+1L(n)an 4 0.
n=1

Remark 1.1. In Lemma 1.4 the assumption that L(°) be non-decreasing

 

is not necessary and can easily be removed by a simple use of Lemma 1.1.
We will, however, have occasion to use Lemma 1.4 only in the case when
L(-) is non-decreasing.

Let us now introduce some notation. Write M = max Sk’

1Sk5n
N = min 8 and R = M - N . The next result we need is due to
k n n n
1SkSn
Heyde [13].

Lemma 1.5. (Heyde)

_I_f_ EIXiIr<D with 1Sr<2 and EXi=u20, then

S

n-l/r(Mn - n u)q‘*' o.

_I_f_ E|xi|r<a with o<r<1 2; Elxi|r<~ with 15r<2

and u < 0, then

n-l/r M 3.3.

-"' O.

n
Finally, we prove the following

Lemma 1.6.
Let Elxilr < 9 with o < r < 2 and write E Xi = u whenever

slxil < 6. Then

3.5.
0)

n-l/r(Rn - n Cr)

where

0 if 0 < r < 1
c={

Inl f l s t < 2.

Proof: We first observe that

N = min 5 = - { max (-3 )}.
n 1sksn k lsksn k '

If 0 < r < 1 and EIXiIr < G then n-1/ran*§' o by Lemma 1.5 and

n-l/rN a;§.

n o by applying Lemma 1.5 with Xi replaced by -Xi. It

-l/r a.s

. . . . r
is immediate that Rn-~ 0 whenever o < r < 1, EIXiI < @.

Next suppose that EIinr‘< a with l S r < 2 and let u 2 o.
, -1/r a.s.
Lemma 1.5 yields n (Mn - n u)-—* o. If we apply Lemma 1.5 once

again with X, replaced by -X, we obtain n-l/r[ max (.8 )}QL§° o
1 1 k
1SkSn
l/r

and it follows that n- (Rn - n u)aé§' 0 whenever 1 S r < 2,
r
Elxil < a and E Xi = u > 0.

Finally, if EIXilr < a with l s r < 2 and n < 0 we obtain

n-1/rM €;§- o and n-I/rc max (-S ) - n(-u)}a‘g' o It readily follows
n k
lSkSn
1/

that n- r(Rn - n|u|)q*§' 0 whenever l S r < 2, EIXiIr < w and
E = < .
Xi u o

This completes the proof of Lemma 1.6.

10

Section 1.3 Main Results.
Theorem 1.1.

For t 2 o the following statements are equivalent:

(a) nt+1L(n)Pr(|X| > nl/r) a o '222_ nl-l/fif rx dF(x) a o.

lxI<n”

1/

(b) ntL(n)Pr(ISnI > n re) r o for all e > o.

1/r

(c) ntL(n)Pr( max IS I > n

1SkSn

k e) r o for 81 e > o.

1Ire) w o for a1 6 > o.

(d) ntL(n)Pr(Rn > n

If t > o the above statements are eguivalent to

(e) ntL(n)Pr(sup k'l/rlskl > e) _. o for all e > o.
k2n

Theorem 1.2.

For t 2 o the following statements are equivalent:

(a) Z ntL(n)Pr(|X| > nl/r) < a and nl-l/fif 1/rx dF(X) a 0.
n=1 |x ([1
'° 1 l/
(b) 2 nt- L(n)Pr(lSnI > n r6) < 9 for a1 5 > 0
n=1
°° l 1/
(c) 2 nt- L(n)Pr( max ISkl > n r5) < a or a1 a > 0
n=1 ISkSn
as

t l/r

(d) E n -1L(n)Pr(R > n
n=1 n

e) < a or a l e > o.

11

If t > o the above statements are equivalent to

-1L(n)Pr(sup k-1/r|S I > e) < a for all e > o.
k “*
n=1 an

Remarks.
1.2 Some particular cases of the above theorems have been given by pre-

vious authors. Spitzer [27] established the (a), (b) equivalence for

the case t o, r = 1, L(x) = 1 of Theorem 1.2 while the case t = l,

r = l, L(x) l was established by stabs [8]. Katz [19] obtained the

(a), (b) equivalence for the r = 1, L(x) = 1 case of Theorem 1.2 (his
Theorem 1) as well as parts of the (a), (b) equivalence for the L(x) = 1
case (his Theorems 2, 3). Parts (a), (b), (e) of our Theorem 1.2
basically summarize Theorems 1, 2, 3 of Baum and Katz [1], [2] unified
and generalized to admit functions L(x) of slow variation different
from the max(1, log x) of Theorem 2 of [2] while Theorem 4 of [2]

is the case r = 1, L(x) = l of parts (a), (b), (e) of our Theorem 1.1.

6

1.3 The condition 2 ntL(n)Pr(|X| > n
n=1 r(t+1)

equivalent to the moment condition EElXI L(IXI)] < a. This

l/r) < w of Theorem 1.2 is

follows by a simple rearrangement of the series (cf. [23, 242]).

1.4 For 1 < r < 2, (a) yields EIXI < a in the case of both theorems
while for r = 1, Ele < a for Theorem 1.2 but not necessarily for
Theorem 1.1. If EIXI < w, the random variables Xi of the theorems

may be replaced by Xi — u where u = E X. The condition

n1-1/rf| |< 1/rx dF(x) * 0 will then reduce to the condition E X = u
x n

(cf. [21, 65]).

12

1-l/rf

1.5 If 0 < r < 1, the condition n /rx dF(x) r o as

1
IxI<n
n r 0 in part (a) of Theorem 1.1 is automatically satisfied. To see

this, it is just necessary to prove that the condition
1—1/r'

x dF(x) w o°
|x|<n1

n Pr(IX| > nl/r) r o as n w 9 implies n

/r
This result follows immediately from Lemma 1.2.
Section 1.4 Proofs.

In this section we construct the proofs of Theorem 1.1 and 1.2

in parallel fashion. We show that

(d)
(a) =) (b) =)(C)

(e)

Equivalence of (a) and (b).

It is convenient to make the proofs for symmetrized random
variables Xi, k = 1,2,3,... and then use the weak symmetrization in-
equalities [23, 245] to transfer to the required results.

. S n S .
Suppose that (b) holds and write Sn Zk=1Xk. FolloWlng Erdbs

[8], write

5 l/r
= >
Ai (Xi n e)
and
n 3
B1 = ( E X, E 0)
j#1,1=1 J
for i = 1,2,3,...,n. For any set E, if E denotes its complement,

we have

13

1he) 2 pr[ u (A n a )3

Pr(S: > n
i=1

[ “[i-l }
= Pr U n (A n B ) n (A. n B.) J
i=lj=l j j 1 1

n i-l

. 2 pr[ n (A n B ) n (A. n 3.)]
i=1 j=1 j j 1 l

n i- 1
a 2n[nAn(AnBJJ
i-l j-l J 1 1
i-l
E E [Pr(Ai 0 B 1) - Pr( U A O A,)]
i=1 j=l j 1
n i-l
(1.4) e 2 Pr(Ai)[Pr(B,) - 2 Pr(A,)]
i=1 1 3:1 3
n
= iflpr(A1)[Pr(Bi) - (i - 1)Pr(Ai)]
E n Pr(Ai)E% - n Pr(Ai)].
Now note that (b) implies n-llrsz E 0. (We will see later that (b)

actually implies n-Ursn B 0.) We only need to prove this for the

o S t < 1 case of Theorem 1.2; the case of Theorem 1.1 and t 2 1
case of Theorem 1.2 being obvious.

n-l/rss

In order to show this we suppose that does not converge

to zero in probability. Then there exists an e > 0 such that either

Pr(nEI/rS: > e) > e or Pr(n1 wllr ‘< -e) > e for infinitely many i.
i ni
For the sake of definiteness assume Pr(n;1/rS 8 >6) > 6 for infinitely
i
many :1. Without loss of generality choose ni+1 > 2 hi. Then for each
j, n < j S 2 n,, we have Pr( 2 x: 2 o) 2-. Since
1 1
k=n +1
i
J j l/r mi 3 ‘1 l/r j s
Pr( 2 Xk > (2 ) e) E Pr( 2 Xk > (2) e). Pr( 2 Xk 2 o)
k=1 k=1 k=ni+1
1 s 1/r
25- >
_ 2 Pr(Sn. e)
1
g.§
2

14
for n, < j S 2 n,, and
1 1

il/r > l/r

Prcs: > (g ) e) 2 Pr(s: a1 a) > e,

i i

we have for ni S j S 2 ni that

l/r ‘5

2 .
e) 2

s
Pr(Sj > (%)

Thus,

2 nt-1L(n)Pr(SS > (2 n)
n=1

nl/r 6)

2n.

9 1
E 2 Z n
i=1n=n

i
6 2n
52 2131‘?)- (oSt<l)

i=1 n=n, n
1

t- nl/r

1L(n)Pr(S: > (5 n) e)

IN

By the weak symmetrization inequalities it follows that

a
2 nt-1L(n)Pr(|SnI > nl/re/zl/r+1

n=1
-l/rss g o
n

) = a, which is a contradiction. This
proves n
We are now in a position to prove the (b) implies (a) part of
the theorems for symmetrized random variables. Indeed, by Lemma 1.3 we
obtain n Pr(Ai) v o as n ~ 0. Thus for 6 > o arbitrarily close to

zero we can choose N so large that for n 2 N,

Pr(S: > nI/re) 2'nQ% - 5) Pr(XS > nl/re).
Repeating the argument with A1 replaced by (-X: > nl/re) and Bi
n
replaced by (- E X? 2 c), we obtain
iii,j=1 J

1/r 1/r

Pr(-8: > n e) 2'nQ% - 5) Pr(-XS > n e)

15

for sufficiently large n. It follows immediately that (b) implies

nt+1L(n)Pr(|Xs| > nllr) * o as n d G in the case of Theorem 1.1 and

a
2 ntL(n)Pr(|XS| > nllr

n=1

) < 9 in the case of Theorem 1.2.

We now have the required information to show that (b) implies

n.1/r8n E 0. Clearly, we only need to prove this for the o S t < 1

part of Theorem 1.2. First let 0 < r < 1. We have shown in the pre-

ceding paragraph that (b) implies EEIXSIr(t+1)L(IXSl)] < a and thus

-l/r P

EIXIr<t+1> < fl. It follows that EIXlr < G, which ensures n Sn w 0.

Now let 1 S r < 2 and suppose that n-l/rSn does not converge to

zero in probability. -We may assume without loss of generality that
E X = p > 0. Then for a given n, o < n < p, we have for sufficiently

large n

l/r l/r

Pr(Sn > n e) 2 Pr(ISn - n ul S n n).

Using the Kolmogorov-Marcinkiewicz law of large numbers it therefore

follows that Pr(lSnl > nl/r

e) w l which contradicts (b). This proves
our assertion.
Since

3 S X

 

 

n v n-1 n-1 l/r n

= < > +—-——,
nl/r (n_1)1/r n nl/r
1/

it follows that n- r med (X) m o. The weak symmetrization inequalities

1
now yield the required result that nt+1L(n)Pr(|X| > n /r) m o in the
u
t
case of Theorem 1.1 and Z n L(n)Pr(IXI > nl/r) < a in the case of
n=1

Theorem 1.2. Finally, we appeal to Lemma 1.3 to obtain the result that
nl-l/r x dF(x) m o as n m 9.
l/r
x.<n

Next suppose that (a) holds. .We note firstly that it is suffi-

l6
1/

r
5) ~ 0 as n m 0

l/r

cient to show that for all e > o, ntL(n)Pr(ISEI > n
O

in the case of Theorem 1.1 and E nt-1L(n)Pr(|S:| > n e)‘< fi in the
n=1

case of Theorem 1.2. This follows again from the weak symmetrization in-

equalities since (a) implies n-UrSn E 0. Note that we have used Lemma

1.4 to obtain this last result in the case of Theorem 1.2.

Firstly, we diSpose of the case r(t+1) < 2. Define

s X: if |xi| < nl/r
K...={O

otherwise,

. s _ n .
and write Snn — 2k=lxin° Since

Pr(ISEI > nllre) =

l/re

Pr(at least one of the lXil's > nl/r, lSzl > n ) +

Pr(none of the IXil's > nl/r, I8:| > nl/re)

l/r l/r

g n pr(|xs| > n ) + Pr(lSEnl > n e),

l/r

it clearly just remains to prove that ntL(n)Pr(|S:nI > n e) m o

a
in the case of Theorem 1.1 and 2 nt-1L(n)Pr(|S:n| > nI/re) < 9 in
n=1

the case of Theorem 1.2. In doing this and Subsequently we shall adopt
the convention that VC denotes all positive constants whose exact values
do not matter. Thus even in a single inequality C can denote different
values and the eXpression l + C S C is a valid inequality using this
notation.

From Markov's inequality [23, 158] and Lemma 1.2 we obtain,

s l/r -2/r s 2
> S
Pr(ISnnl n e) c n E(Snn)

_ l-2/r s 2
(1.5) - C n E(an)

o(n't[L(n)J'1>,

17

which completes the Theorem 1.1 part-of the proof. For the Theorem 1.2

case we return to (1.5) and see that

a
Z nt.1L(n)Pr(ISS I > nllrc)
nn
n=1
' -2/r 2
S C 2 nt L(n) I le dPr(XSS x)
l/r
n=1 x (n
' n
s c 2: nt'z/run) 2 kzerr(k-l s |xs|r< k)
n=1 k=1
9" .

(1.6) = C E kZ/rPr(k-l S lxs|r < k) 2 nt-2/rL(n).

k=l n=k
Now,

°° 2 2
2 nt- /rL(n) S C In xt- lrL(x)dx
k
n=k
= C kt-Z/r+1L(k)fi L(xk)[L(k)]-1xt-2/rdx

By Lemma 1.1 we have

L(xk) a(xk) -.l . xk a(u2d
L(k) =a(k) x EXPUk u dub

where a(u) * l as u m fi. Therefore, given 5 > 0 however small we

can choose k so large that for all x 2 l

#:(f; < -— xexp{(1+¢s)f‘k in“

6
= (l+6)x .
An appeal to the Lebesgue dominated convergence theorem then yields

I: L(xk)[L(k)]-1x t 2/rdx “'I: xt-2/rdx.< 0,

as k m a, and hence

18

a
z nt'Z/rL(n) = 0(kt'2/r+11(k)).
n=k
Returning to (1.6), we then have
° 1 1
2 nt- L(n)Pr(ISEnI > n /r€)
n=1
° +1
s c z kt L(k)Pr(k-l s |x3|r < k)
k=l
m t s l/r
s c 2 k L(k)Pr(IX I >k )
k=l

which is finite using (a) and the weak symmetrization inequalities.

We next consider the case r(t+1) 2 2. We must have t > 0 so

t+ +2 r

< V o
max(2(t+1) , 2) V < l. Def1ne

take

8 s if I s| <
lam-{)1 1%.

0 otherwise,

n
. S 3 .
and write S = 2 Xk . Wr1te also
nn n

An = (Ix:|> 1%n1/r for at least one k S n),
B = (IXE' > nv/r for at least two k's S n),
n
_ 3 l. l/r
Cn - (Isnn| > 2 n )'

We can take a 1 without loss of generality. We assert that

(lssl >n1/r)CA U B UC.
n n n n

, - s 3
To see th1s, let w E Bn' Then th(w) # Xk(w) for at most one value
of k. If w is also in in then for all values of k we have

l/r

lxiam 5%- n ; this holds in particular for that value of k (if

. s s - -
any) for which th(w) # Xkcn). Thus for w 6 An N Bn the sums

l9

s s . l l/r — .
Sn(w) and Snn(w) differ by at most 2 n . If m E CD also, (1.e.
l/r /r and

- - - s l s l
w 6 Ala fl 1311 0 Cu) then Isnn(w)I s 2 n so that Isn(w)l s n

w E (IS:(w)I S nl/r). It follows that

s 1/r - - -
S 2
(ISnI n ) An 0 BH fl Cn’

and thus our assertion is proved. It follows that

l/r

(1-7) Pr(Is:I > n )

S Pr(An) + Pr(Bn) + Pr(Cn)

s nPr(IXSI >11- nI/r) + n2[rr(IxSI > nv/r)]2 +
+ Pr(ISEnI >-% nl/r).

It is then clear, using the weak symmetrization inequalities and Lemma

1.4, that the cases of both theorems will be proven if we can show that

l/r

when nt+1L(n)Pr(IXSI > n ) ~ 0 as n H a,

nt+1L(n)[Pr(IXSI > nV/r)]2 < a
1

IIM

n

and

nt-1L(n)Pr(IS:nI >‘% nl/r) = 0(n-T)

where T > 1. This is what we propose to do next.

l/r

, t+l s .
S1nce n L(n)Pr(IX I > n ) d o as n ~ w, we can, given an

arbitrarily small 6 > 0, choose a constant A so large that

xr<t+1)L(xr)qs(x) < 6 for x 2 A where qs(x) = Pr(IXSI > x). Then,

for n > Ar/V,

nv(t+l)L(nv)qs(nv/r) < 6,

20

so that

(t+1)(1-2V) -2 62

+1L(n)[Pr(IXSI > nV/‘nz < n L(n Km )1

and

v/r

+1L(n)[Pr(IXSI > n )]2 < on

"M

n 1

since (t+l)(l-2v) < -1.
Let A be an even integer greater than 2 rt(2-r)-1. From

Markov's inequality we obtain

l/r

l
Pr(ISEnI >'§ n )

s C n-A/rE(ss )A
nn

-A/r

SCn {n E(X: HA) +n(n-1)E(X:nA)-2E(X:n)2+...}.

Since there are only a finite number of terms in the preceding expansion
(the number depending on A but not on n) it will be sufficient to
show that each term goes to zero at an 'appropriate rate'. Let

(211,212
A bound for the corresponding term in the preceding expansion is given

21
m‘-A/rE(X1n) 1 E(X§n)21m} NoW‘from Lemma 1.2 we see that the

'21
factors E(X:n) j are bounded for 2ij < r(t+1), a
v[2i /r-t-1J

,...,2im) be a partition of A into positive even integers.

by C n

o(n j [L(nv)]-1) for 21j > r(t+1) and are o(nn[L(nv)J-1)
for 2ij = r(t+1). Consequently, if u = Z 2ij, v = number of
21j>r(t+l)
ij =-% r(t+1) and w = number of ij >-% r(t+1), we see that
21
um n) I...E(X:n) m

21

= 0(nmrA/r+NV+vu/r-v(t+l)w[L(nv)J-(v+w))

o(n_B),

where B = min(-% A +-%A, -1 +% - A); + v(t+l)). This readily follows

from the inequalities l - V(t+1) < o, 1 - 2\)/r‘< 0. Further, since

A > 2 rt(2-r)-1 (> r(t+1)) implies t - l - B < -1 it follows that

nt-1L(U)Pr(ls:nl > .21.”. Ill/r) = 0(n'T)’

where T > 1.

The equivalence of conditions (a) and (b) is thus established

for both theorems.

Equivalence of (b) and (c).

By Levy's inequality [23, 247], we have

Pr( max IS - med(s - S )I > nI/re) § 2 Pr(IS I > nllre).
k k n n
lSkSn
_ , , -l/r P
Also, in the case of both Theorems 1.1 and 1.2 (b) 1mpl1es n S m o

n

-l/r

which yields n (Sn - sk) E o and hence med{n-l/r

(Sn - Sk)} m 0.
Thus we see that (b) and (c) are equivalent; the (c) implies (b) result

being obvious.

Equivalence of (b) and (d).

Suppose that (b) holds. This implies that (c) holds. Now

R = max S - {- max (-S )}
“ lSkSn “ lSkSn k

S 2 max I3

I
lSkSn k

and it follows that

1kg) S Pr( max IS I > nl/re/Z).

Pr(Rn > n
lSkSn

k

22

Thus (d) holds for both the theorems. As for the (d) implies (b) part

of both the theorems, we observe that

R = max Sk - min Sk
“ lSkSn lSkSn
2 -
Sn min Sk
= S + max (-Sk)
lSkSn
2 Sn + (-X1)
n
= 2 .
k=2 xk
Similarly
R 2 X - min 3
n l lSkSn k
= X + max (-S )
1 lSkSn k
2 -
x1 811
n
="Z .
k=2 xk
Since X1,X2,... are identically distributed it follows that

l/r l/r
> Z >
2Pr(Rn n e) _ Pr(ISn_1I n 5).

This completes the proof of the equivalence of (b) and (d).

Equivalence of (b) and (e).
We first observe that it is only necessary to show that (b)

implies (e) as the (e) implies (b) part is trivial.

For 21 S n,< 21+1, we have

Pr(sup k-llrISEI > e) E Pr(supi k.“r

Isil > e>
k2n k22

23

G l/r s
E 2 Pr( max m ISmI > 6)
i=1 szm<2j+1

S

IIA

(1-3) X Pr(IS

o/r
2J'+1I > ZJ E)’
i=1

where we have used Levy's inequality to obtain the last inequality.

Thus in the case of Theorem 1.1,

ntL(n)Pr(sup k-l/rIS:I > e)

an
S 2 2 2<i+1>tL(Zi+1)Pr(ISS.+1I > 2j/re)
j=i 2]
s 2 I; r2(j+1)tL(zj+1)Pr(|ss 1| > 2(j+1)/r e z-l/r)] 2-(j-i)t
- ,+ _ 7
j=i 2.]

t s 1 r .
and since n L(n)Pr(ISnI > n / e) d o as n d a (uS1ng the weak
symmetrization inequalities) we can, given arbitrary 5 > 0, choose an

I such that

-l/r

21tL(2i)Pr(ISSiI > zl/r.e.2 ) < 6 for all i > 1.

2
Therefore for n > 21,

l

1“Isl“:I > e) s 25.2"(2t - 1)’ ,

ntL(n)Pr(sup k-
k2n

that is,

ntL(n)Pr(sup k-Ur

ISEI > 6) ~ 0 as n m'w.
an

For the case of Theorem 1.2 we return to (1.8) and can write

t"1L(n)Pr(sup k-l/rISEI > 6)

n=1 k2n

24
i+1

w 2 '1 t-l -1/
= 2 n L(n)Pr(SUp k rIsil > 6)
i=0 _ i an
n—2
a o . 1 a o
s c 2 21t1(21+ ) 2 ads8 +1| > zJ/re)
i=0 j=i Zj
a ./ j .t .+1
= c 2 Pr(ISs. | > 2J re) 2 21 1(21 )
. 3+1 .
320 2 1=o
'3 '+1) '+1 8 '
s c 2 2‘J t1(2J )Pr(I3 I > 23”..)
j+l
j=o 2
a t / 1
s c 2 2j L(2j)Pr(ISst > 2j re.2' /r)
j=o 2
¢ Zj+1- .( 1) ./ 1
(1.9) s c 2 2 2J t‘ L(2j)Pr(ISS | > 2J rc.2' /r).
j=o J 23
n=2
Now for 2j S n < 2j+1:
(1.10) Pr(Is:| > nI/re) 2-% Pr(ISSjI > 2<j+1)/re)
2

since

Pr(s: > 2(3+1)/r€) e Pr(Ssj > 2<j+1)/re xS +...+ x: 2 o)

)
2 23+1

2'l Pr(SS > 2(j+1)/re).
2 21

Therefore, using (1.10) in (1.9), we obtain

a
2 nt-1L(n)Pr(sup k-l/rISSI > 6)
n=1 an k

E C 2 Z nt-1L(n)Pr(ISzI > nI/re.2-2/r)

t

25

which is finite using condition (b) and the weak symmetrization in-
equalities.

-1/rs P

Finally, since (b) implies n n ~ 0 (in the case of both
1/

theorems) which in turn implies med(n- rSn) ~ 0, it follows from the

weak symmetrization inequalities that

ntL(n)Pr(sup k-llrISkI > e) w o as n ~'¢
k2n

in the case of Theorem 1.1 and

2 n -1L(n)Pr(sup k-l/rISkI > e) < a for all e > 0
n=1 an

t

in the case of Theorem 1.2.

This completes the proof of both the theorems.

Remarks.

.Lzé In the proof of the equivalence of (b) and (e) we have not used

the fact that L(n) is a function of slow variation. Indeed, it could
be replaced by an arbitrary non-negative, non-decreasing function of n.
Similarly in the (b), (c) and (b), (d) equivalence parts of both theorems
we may replace the coefficient sequence by a sequence {an} of positive

l/r

real numbers with an 2 c > o and n by any positive sequence of
real numbers {Kn} where xn m'm. Summarizing we have the following

result:

Let IX“: n 2 1} .23 5 sequence of independent, identically

distributed random variables and {xn}, {an} ‘23 sequences of
2 > #6

positive real numbers with an c o for al n and xn

as. n “’G.

Then

26

(a) the following statements are equivalent:

0 > .4 >
1) anPr(ISnI xne) o for all e 0.
ii) anPr( max ISkI > xne) fl 0 for al e > o.
lSkSn
iii) a Pr(R > x 9) ~ 0 for all e > o
n n n ----

t .
If 3n = n f(n) (t 2 o) where f(n) 18'32 arbitrary non-negative,

l r
non-decreasing function g; n and xn = n / , the above state-

ments are equivalent.£g

iv) ntf(n)Pr(sup k-l/rISkI > 5) ~ 0 for a l e > o.
k2n

(b) The following statements are equivalent:
O

. > < > .
1) 2 an Pr(ISnI xne) a for al 6 0

n=1

H

0
V
0

ii) 2 anPr( max Iskl > xne) < a for al
n lSkSn

... > < a >
111) 2 anPr(Rn xne) for a1 6 o.

n

f an = nt'lfin) (t > o), f J defined 3(a), E x = nl/r,

-— n

the above statements are equivalent to

iv) 2 a f(n)Pr(sup k-I/rIS I > e) < a for all e > o.
n k -——--——-
n k2n

1.7 The case t = o of the (a), (b) and (e) equivalence parts of the
theorems is worth further mention. Baum and Katz [2] (Theorem 2) have,
in.effect, established by methods quite similar to those used here, the

gqUiva lence of

O

(a) 2 log n Pr(IXI > nI/r) < a and “1-1/r. 1
n=1 IxI<n
" 1 1/

(b) E n- log n Pr(ISnI > n r
n=1

/rx dF(x) m o.

e) < a for all e > o.

27

D
(c) 2 n-1 Pr(sup k-1/rIS I > e) < ° for a1 6 > o.
k ~—
n=l k2n

It is interesting to note that the logarithm (a function of slow
variation) dr0ps from (c). Because of technicalities in the proof this
result is not amenable to generalization in the direction of replacing
the logarithm by a general function of slow variation. As one would
expect, there is a complementary result for convergence to zero (along
the lines of Theorem 1.1). This follows because the condition

Pr(sup k-llrISkI > e) w o as n ~'¢ for all e > o is precisely the

an

condition n-1/ran*§'

0. -We immediately obtain, making use of Theorem

1.2, the equivalence of
(a) E|x|r < a and E x = 0 ‘lg 1 s r < 2.

1”e) < a for all e > o.

a
(b) zn'lPr(|Sn| > n
n=1‘

(C) Pr(sup k-l/rISkI > e) w o ‘igr all e > o.

k2n
Section 1.5 Examples.

In this section we give some simple examples to demonstrate the
significance of having the complementary forms of Theorems 1.1 and 1.2.
These examples illustrate that neither theorem will give consistently
better rates for convergence in the law of large numbers for specified
generating type of random variable.

Suppose that for t 2 o,

Pr(X = n) = C n-t-2(log n)-1, n = i;2, : 3,...

We have

28

Pr(IXI > n) ~ 2C I: x-t-2(log x)-1dx

2 C -t-l -1
" t+1 n (log n) ’

n

so that in particular nt+1(log n) Pr(IXI > n) w o as n m 0 for any
0 < n < 1. Theorem 1.1 therefore implies that nt(log n)nPr(ISnI > ne) w o
as n ” a for all e > o and any 0 < n < 1. On the other hand, in
order to make as strong a conclusion from Theorem 1.2 we would have to

a
have 2 ntPr(IXI > n) < Q and this is not true.

n=1
Alternatively, suppose that for t 2 o,

Pr(X = n) = C n_t-2(log n)-2, n = :;2, + 3,...

We have

Pr(IXI > n) ~ 2C I: x—t-2(log x)-2dx

~ -—- n-t-1(log n)-2:

t+1 <

and in particular we shall have EIXI fl so that Theorem 1.2

Q
yields the result 2 nt-lPr(I3nI > ne) < ° for 311 E > 0- 0n the
m=l
other hand, to make use of Theorem 1.1 we need a result of the form
t+1 . .
n L(n)Pr(IXI > n) w o as n 4 a for a non-negative, non-decrea31ng
function of slow variation L. In our case L(n) = o(log n)2. We can

then conclude that ntL(n)Pr(ISnI > ne) * o as n w m, for all e > o,

a
which is a weaker result than 2 nt-1Pr(ISnI > ne) < O.
n=1

Section 1.6 Some Results on One-sided Convergence Rates.
We preserve the basic set up of section 1.1. A simple application

of the strong law of large numbers shows that if EIXI < a and E X > o

29

then Pr(Sn S x) m o as n ~'¢ for all x, -G < x.< 9. In this
section we use the results of section 1.3 to yield information on the
rate of convergence of the sequence {Pr(Sn S x): n 2 1} to zero for
all x, -"< x'< a. We also state some results on the rate of con-

1fix), 0 < r < 2} to zero for

vergence of the sequence {Pr(Sn S n
all x, -@ < xl< 0.
Let X' = min (o,X) and X = X+ + X-. ‘We establish the

following theorems.

Theorem 1.3.

Suppose EIXI < a and E X > 0. Then 12 order that for some

 

it is necessary and sufficient that

nt+1L(n)Pr(IX-I > n) w 0 .EE n ~.m,

Theorem 1.4.
Suppose EIXI < a and E X > o. ALnecessary and sufficient

condition for the convergence pf the series

Q
2 nt-1L(n)Pr(Sn 5 x), -O < x < co,
n=1
for some t 2 o ‘13 that EEIX'It+1L(IX'I)] < a.

 

Some immediate remarks are in order.

Remarks.
1.8 The L(x) = l, t 2 1 case of Theorem 1.4 has been obtained by

Heyde in [12].

30

1.9 For the sake of completeness we state the following result on the

exponential convergence rate of Pr(Sn S x) which is due to Heyde [12].

Suppose that EIXI < G and E X > 0. Then ip_order that for

 

some t > o

a
(1.11) zlempasn s x) < a, -a < x < a,
n:

ri£n£§ necessary and sufficient that X“ has an analytic char-

 

acteristic function.
(The term "analytic characteristic function" is used for a characteristic
function which is analytic in a strip containing the origin as an in-
terior point.)

We remark that condition (1.11) may be replaced by an equivalent

condition:
tn
e Pr(S S x) w o as n m a, -m‘< x < a,
n

1.10 It is clear that analogous results will hold in the case E X < o.
The proofs of Theorems 1.3 and 1.4 are obtained from the follow-
ing lemmas which we now state and prove. The lemmas are constructed in

the fashion of Heyde [12].

Lemma 1.7.

La; z|x|<~ 222 “>0 9__1.§£ E|x+| =~ 2m EIX'|<~.
31133

(a) nt+1L(n)Pr(Ix'| >n)—»o _a_s n-*°°

for some t 2 0 implies

ntL(n)Pr(Sn S x) -0 0) —Q < X < O;

31

lt+l

(b) EEIX' L(IX‘|)] < a

for some t 2 0 implies

a
Z nt-1L(n)Pr(Sn S x) < 9, -9 < x < '.

n=1

Lemma 1.8.

Let EIXI < a and E x > 0. Then

(a) ntL(n)Pr(Sn S x) m o .as n fl @, -¢ < x < a,

 

or some t 2 0 implies

nt+1L(n)Pr(IX'I > n) w o ‘g§_ n d 6";

t.1L(n)Pr(Sn S x)‘< ¢, -¢ < x.< 0,

for some t 2 0 implies

 

EEIx‘|t+1L(|X‘|)] < s.

Proof of Lemma 1.7: Following Heyde [12], define a new random

variable Y as follows

* (X if X'< K
0 otherwise,
where K(> o) is a constant chosen so that E Y > 0. Then Y S X
and we have nt+1L(n)Pr(IYI > n) w o as n ~ w in the case of (a) and

|t+1

EEIY L(IYI)] <1a in the case of (b). It follows immediately from

Theorems 1.1 and 1.2 that

ntL(n)Pr(Y1 + Y +...+ Yn S n E Y - ne) d o

2

32

for all e > o in the case of (a) and

a
2 nt-1L(n)Pr(Y1 + Y

+...+Y SnEY-n€)<a
1 n

2
n

for all e > o in the case of (b). Now for sufficiently large n

and choosing a so small that o < e < E Y we have

Pr(Y1 + Y +...+ Yn S x) S Pr(Y1 + Y +...+ Yn S n(E Y - 3))

2 2

and since

S E S
Pr(Y1 + Y +...+ Yn x) Pr(Sn x)

2

both (a) and (b) follow immediately. The proof of the lemma is now

complete.

Proof of Lemma 1.8: Write Yi = Xi - n where E Xi = p and
n
2 = 2 Y, - S - nu. Then
n i=1 1 n

S = S - .2. S-
Pr(Sn x) Pr(Zn x nu) Pr(Zn nc)

for c > p and n sufficiently large.
Since EIXI < 9, it follows by a simple rearrangement (cf. [23,

an
242]) that for arbitrary e > 0 we have 2 Pr(Yi < -(c+e)n) < ”.

n=1
Also, since the terms in this series are non-increasing, it follows
frothemma-l.4 that
(1.12) nPr(Yi < -(c+€)n) m o as n d 0.

-Write

A1 = {Yi < —n(c+e)}

33

and
n
Bi = [I ‘8 Y.I < (n-1)e}
3:1:j*i
for i = l,2,...,n. Then, we have

n
pr(zn 5 -nc) 2 PrEiLilmi 0 Bi)]

n i-l
2 z Pr(Ai)[Pr(B.) - 2 Pr(A.)J,
i=1 1 j=1 J

as in (1.4). Thus,

n
Pr(z 5 -nc) 2 Z
n

i=1Pr(Ai)[Pr(Bi) - n Pr(Ai)].

Let 0 < n < 1 be arbitrary and 6 > 0 such that l - 26 2 n. It

follows from the weak law of large numbers that we can find an integer

N1 such that

Pr(Bi) > 1 - 6 for n 2 N1.

Also, from (1.12) we can find an integer N2 such that

< 2 .
n Pr(Ai) 6 for n N2
Thus, for n 2 max (N1,N2), we have
S 2 S-
Pr(Sn x) Pr(Zn RC)
- 2 nn Pr(Yi < -(c+€)n).

Thus we obtain

nt+1L(n)Pr(Yi < -(c+€)n) 4 o as n d a

34

in the case of (a) and

Z ntL(n)Pr(Yi < -(c+e)n) < a
n=1

in the case of (b). We now introduce the random variable U = Y' and

obtain
nt+1L(n)Pr(U < -(c+e)n) 4 o as n d a
in the case of (a) and
a t
2 n L(n)Pr(U < -(c+€)n) < a
n=1
in the case of (b). It follows that
nt+1L(n)Pr(IX'I > n) w o as n 4 a
in the case of (a) and

EEIX'|t+1L(|X'|)J < a

in the case of (b).
This completes the proof of Lemma 1.8 and hence of the Theorems

1.3 and 1.4.

Remark 1.11. Let EIXIr<° with o<r<2 and EX=o whenever

 

l S r < 2. It follows from the Kolmogorov-Marcinkiewicz strong law of

1/

large numbers that Pr(Sn S n rx) m o as n ~ 6 for all x,

-¢ < x‘< o. The methods of this section easily yield the following

1lrx)} to zero for

result on the rate of convergence of {Pr(Sn S n
-9 < x‘< 0.

Let EIXIr < a with o < r < 2 and suppose that E X = o ‘if

 

35

l S r < 2. Then

1/r

t
(a) n L(n)Pr(Sn S n x) w o, -a < x < o,

for some t 2 o “if and only if

- l
nt+1L(n)Pr(IX I > n /r) * o;

and
° t 1 1

(b) 2 n - L(n)Pr(Sn S n /rx) < 0, -¢ < xw< 0,
n=1

_'_f and only __'_f_

-|r(t+l)

EEIx L(IX'I)J < “;

for some t 2 o.
It is clear that analogous results hold for the convergence of

the sequence {Pr(Sn 2 nl/rx)} to zero, for all x, o < x < a.

Section 1.7 Miscellany.

In this section we give some related results of a miscellaneous
nature which are obtained as corollaries to the theorems of section 1.3.
Our first result concerns the convergence of a series of alternating

positive and negative terms.

 

Theorem 1.5.
Let EEIXIr(t+1)L(IXI)] < 9 for some t > o with r(t+1) 2 1

and write E Xk = n. Then the series

a
2 (-l)nntL(n)Pr(sup k’l/rls - kuI > e)
k
n=1 k2n

36

converges for all e > 0.

Remark 1.12. For r = 1 and L(x) = l we get the corollary to Theorem

3 of Baum and Katz [2].

Proof: The procedure is to demonstrate that the tail of the
series in question can be made arbitrarily small. This method was also
used by Baum and Katz [2].

Let

3n = Pr(sup k-l/r

ISk - kuI > e}
an

for arbitrary e > o, and let

Tn’k = {ntL(n)an - (n+l)tL(n+l)an+1j-_...j-_(n+k)tL(n+k)an+kI.

Since {an} is a non-increasing sequence of real numbers, T is

n,k
minimized if
a

= a a = a c.
n n+1’ n+2 n+3’ et

Thus, if k = 2m+l, m =-E%l is a non-negative integer and we have

Tn,k E [anEntL(n) - (n+1)tL(n+l)]+...+

+ an+2m[(n+2m)tL(n+2m) - (n+2m+l)tL(n+2m+1)]}

= {:0 an+2,[(n+2t) L(Iger’L) 7 (“242“) ““2““

(k-1)/2

= 2.2 (“+2"+1’t“‘“+2‘)an+2t[<1 - Erin” - W3

=o
(k'1)/2 t-l
2 -Ct 2 (n+26+1) L(n+2L)an+2,.

L=o

37

Similarly if k is even

(k/2)-1 t_1
T 2 - Ct Z (n+2L+1) L(n+2L)a

1'1, k {’30 n+2L .

On the other hand

Tn,k = {ntL(n)an + [-(n+l)tL(n+l)an+Et..2:(n+k)tL(n+k)an+k]}

is maximized if

3n+1 = an+2’ an+3 = an+4’ etc.
and therefore, as before, we have
t (k/Z) t-l
n L(n)a + Ct Z (n+2L) L(n+2£)a if k is even,
. n L=l n+ZL
s
Tn,k l t (k-1)/2 t_1
n L(n)a + Ct 2 (n+ZL) L(n+2£)a if k is odd.
n L=1 n+2£

We now appeal to Theorem 1.2 to obtain

a
z nt-1L(n)a < a
[1
n=1

and to Lemma 1.4 to yield
t

n L(n)an * o as n w 9.

The proof of Theorem 1.5 is now complete.

Remark 1.13. The conclusion in Theorem 1.5 remains true if L(‘) is
a non-increasing function and for some t > o and o < r < 2
E[|x|r(t+1)L(|xI)] < a and EIXI < a.

Next we state a simple corollary to Theorems 1.1 and 1.2 which

yields some information on the rate of convergence of the sequence

38

[n-Ilr max (Sk - kp)} to zero (see Lemma 1.5).

lSkSn

(a) For t 2 o, nt+1L(n)Pr(IXI > nl/r) m o and

l-l/r q ,
n I l/rx dF(x) 0 imply
x.<n

ntL(n)Pr(I max SkI > nllre) * 0 or al 3 > o.
lSkSn
at 1/
(b) For t 2 o, 2 n L(n)Pr(IXI > n r) < a and
n=1
111-1/r x dF(x) * 0 imply
l/r
IxI<n
”t1 1/
2 n - L(n)Pr(I max Skl > n re) < a for a1 6 > 0.
n=1 lSkSn

Finally we state as an application of the theorems of section 1.3
a result on the rate of convergence of maximum likelihood estimates
which strengthens the work of Brillinger [4]. Let 6n be the maximum
likelihood estimate based on n observations and 60 be the true value
of the parameter. Suppose that in addition to the eight assumptions of
Wald [29], which guarantee the consistency of maximum likelihood estimate,

the following three assunptions of Brillinger [4] are satisfied:

(1) For each 6, there exists 0(9) such that for o < p S 0(9)
* v
EIlog f (x,9,p)I < a for some V > 1.
(ii) There exists r with o < r < a such that
*
EIlog cp <x.r>|v < ~-

(iii) EIlog f(x,eo)lv < a.

39

(The notation used here is that of Wald [29].)
Brillinger showed that if Y and e are any positive numbers, then

there exists no such that
- 2 s - ' s
(1.13) Pr(Ien eol e) v/nvl 1f 1 v<2
< k/ %v if v 2 2
n

for all n > no and some positive constant k.
Using the sharper results of section 1.3 it is easily seen that

under the assumptions of Wald and Brillinger we have for v 2 1,

(a) nV-lPr(I6n - BOI 2 e) 4 O for all e > o,

a
(b) 2 nv-zPr(I8n - eol 2 e) < a for all e > 0.
n=1

CHAPTER II
CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS

FOR INDEPENDENT RANDOM VARIABLES

Section 2.1 Introduction.
Let IX“: n 2 1} be a sequence of independent, but not nec-

n
essarily identically distributed, random variables and write 8n = 2 Xk.
k=1

Take 0 < r < 2. Suppose that the random variables Xn are uniformly

bounded by a random variable X in the sense that
Pr(IX I > x) S Pr(IXI > x), all x > o.
n

Set qn(x) = Pr(IXnI > x) and q(x) = Pr(IXI > x).
Then, if qn S q and EIXIr < a with r < 2 ‘32 have (see [23,

242])

n-l/r ; (Xk - a )€;§- o
)
k=l k

where

{0 i-_f_ r<1,
a =
k E )1k ;3_ r 2 1.

In this Chapter we study the rate of convergence of the sequences

n l/r -1/r k
{Pr(l 2 (xk - ak)I > n e): n 2 1}, {Pr(sup k I z (x, - a,)I > e): n 2 1}
k=1 an j=l J J

l/re): n 2 1} to zero for arbitrary

and {Pr( max I 2 (X

-a)>
15kg. 1.1 J J ' “

e > o. The results obtained here are in the spirit of those obtained
in Chapter I; we concern ourselves with the modifications necessary when

the condition of identical distributions is replaced by this one of

40

41

uniform boundedness. We also generalise a reSUlt of Baum, Katz and
Read [3] on exponential convergence rate for the sequence
{Pr(ISnI > amen.
In section 2.2 we prove some preliminary results on the a.s.
convergence of the sequence {n-1/r(Sn - E Sn)} to zero. We also
state a lemma which we shall use in the later sections. In section 2.3
we state the main theorems of this Chapter; these are proved in section
2.4. (Section 2.5 is devoted to the study of exponential convergence

l/r

rates for the sequence {Pr(ISnI > n 5)}. Finally, in section 2.6

we give some miscellaneous related results.

Section 2.2 Some Preliminary Results.
Let {Xhz n 2 1} be independent, but not necessarily identically
distributed, random variables and take' 0 < r < 2. We first prove the

following

Theorem 2.1.

l/r

(a) 2 n-lPrIISn - med(Sn)I > n e} < a

n=1
for all e > 0 implies

n-UrESn - med(Sn)]q*§' o.

(b) If in addition IX,Ir < i, then
__.__..________ 1 _____

l/re} < fl for all e > o

a
Z n-lPrIIS - E S I > n
n=1 n n

implies

-l/r a.s.
n (Sn ~ E Sn) 0.

42

Proof: By weak symmetrization inequalities and the hypothesis

of (a) we have

a
O > Z n-IPrIISn - med(sn)I > nl/re}

n=1
1 ° 1 1/
2'2 E n- Pr(ISsI > n r(6/2)}
n
n=1
‘+1
a 21 -1
..2} z z n’lrrIIs:| >n1/r(€/2)}
i=0 “:21

231 z 2-(i+1) z PrI‘ISEI > 2(i+1)/r(€/2.)}
i=0 n

a o o a
2 2-(1+1). 21PrIISSiI > 2(1+1)/r(€/2)}
i=0 2

J-‘It—I

(i+1)/r

O
=.% z Pr(IssiI > 2 (5/2)}-

i=o 2

Now

pr{2'j/rssj > 21/r(e/2)}

2 PrIZ-j/r(S . - S . ) > 21/r(€/2),SS. 2 o]

2.] 2.]-1 2]-]-

1/

‘l -j/r s s r
2 2 Pr{2 (S - S j-l) > 2 (6/2)}:

and we therefore have

.1. 2 (i+1)/r

2 Pr(IssiI > (6/2)}

i=0 2

a)

l 1/

a '/
2-—- 2 2:12’1 rIss. - 38. I > 2
21 21-1

16 r(6/2)}.
i=1

Consequently, by the a.s. stability criterion [23, 252] we have

n—l/r s 323-

n

S o, and hence n-l/rESn - med(Sn)JQL§' o.

43

Under the hypothesis of (b), as in (a) we conclude that

n-l/rSs Q;§- o. If f8 is the characteristic function of K: and g3
n k n

the characteristic function of n-l/rsz, then it follows that
s n s -1/r
gn(u) = H fk(n u) w l uniformly in every finite interval.
k=l

Now we have (see [23, 244])

Imed(Sn) - E snl s./2 var(Sn)

 

=‘/’ n
2 2 var( )
k=1 xk

 

= /'TT 3
2 var( )
k=l xk

Also, Ixil < 11" implies Ixil s 2 il/r, and thus |x:| s 2 nl/r.

Therefore, by the truncation inequality [23, 196] we have

“ -2/r s n 2
2 n var(X ) s z‘f x dF.(x)
. 1. 1
i=1 i=1 lesz
n
S -12 2 log Re f:(l/2n1/r)
i=1

= -12 log g:(l/2) d o

as n ~ Q. It follows that

-l/r q
n lmed(Sn) - E Sn 0,

and hence that n-l/r(Sn - E Sn)q‘§° 0.

Remarks.

2.1 The r = 1 case of Theorem 2.1 has been obtained by Baum and Katz

[2] using the same methods.

2.2 The converse of (a) is not true. For, if we let X1 be a symmetric

44

random variable such that
Pr(X1 > t) = Pr(X1 < -t) = (log t)'1

for large t and define X2 = X3 = ...... 3 0 then n—

' 1 1/
but 2 n- Prfn- rlsnl > 1} = a.
n=1

2,; In (b), the growth restrictions on the Xi's are used to rid us
of the ‘med(n-1/rSn) terms. We cannot avoid some assumption of this
type. In this connection see [23.

In part (i) of Theorem 2.2 below we obtain a criterion for the
strong law which is stronger (for r S 1) than the classical extended

criterion due to Brunk [5]. In part (ii) of the theorem we generalize

a recent result due to Baum and Katz [2].

Theorem 2.2.

Let {sz k 2 1} [be a sequence of independent random variables

Q
with E xk = o, k = 1,2,... and 2 ks'l'zs/rslxkl28 < a or
=1
some 8 2 l, oI< r < 2. Then
(i) n-l/rS €;§- o,
n
a 1 1/
(ii) 2 n- PrflSnI > n re} < 0 for a l e > 0.
n=1
Remarks.
0
2.4 Brunk [5] showed that if E Xk = o and 2 k-(S+1)/rEIXk|ZS < a

k=l

-l r a.s.
for some positive integer s then n / Sn‘- 0. Thus for r s 1

the result in part (i) of Theorem 2.2 is stronger than Brunk's result.

2.5 The r = 1 case for part (ii) of Theorem 2.2 is due to Baum and

Katz [2].

45

Proof of Theorem 2.2: The s = 1 case for (i) is the well
known Kolmogorov criterion for the strong law of large numbers [23, 253].

For 3 > 1 we use the methods of [23, 259] and define

x-——-—
* x if |X|<n S
n n
xn={

0 otherwise.
Then

- * -
n 1“1an = 0(log log 1nllr)

ns-l-ZS/r s-l-ZS/r

2

n

EIXZlZSs 2: n I-:|xn|ZS < a,
n

and by Chebyshev's inequality,

*
r21”. Pr(Xn ,4 xn) = EpulxnI 2 n )

s 2 ns-l-ZS/rElxn|23 < a.
n

Thus, the sequences X.n and x: are tail-equivalent [23, 233], and
it suffices to prove that the assertion holds for random variables

Kn which satisfy the assumptions of the a.s. stability criterion in
terms of variances [23, 258]. For any subsequence n S n S ... of

l 2

[n] we have,

t25 = {n-Z/r 2 E X2}S
k k n <nSn n
k-l k
-23/r s
S
nk E(Z Xi)
n
k
- -2
511:1 8/]: 2 I-zlxl23
+1 n
nk-l

46

“k
s 2 ns-l-Zs/r EIthZS
nk_1+l
° 2
and it follows that 2 tks < “. -Since for any sufficiently large k
k=l a
exp(-e/2 ti) < tis for e > o it follows that 2 exp(- 6/2 t i) < a
k=l
which implies [23, 258] that n-llrsnfiég'
In the case of (ii) we have for s 2 1,
° 1 1/ ° 1 2 2
2 n- Pr(IS I > n re] S C 2 n- n- s/r EIS I S
n n
n=1 n=1
° -22
S C Zn 8- s/r k5 1EkaIZS
n=1
° 12 2
so 2k8--8/rEkaIs
k=l
<09

where we have used [23, p. 263, problems 4 and 5] to obtain the second
inequality.
Finally, we state as a lemma a classical result [23, 317] which

we will have occasion to use frequently.
Lemma 2.1. (Degenerate Convergence Criterion)
: 2 . .
Let {xh n l} 522 a sequence of independent random variables
' - . '< = S .
and write Sn 2 Xk Let 0 r < 2 and set Fk(x) Pr(Xk x)

. k=l
-llrSn g 0 __£ and only 'f or al 6 > o as n w’“

Then n

(2.1) 2 Pr(I I > 111 re) d o
k=l xk

n
n-l/r x dFk(x) * o

(2.2)
k=l IxI<n”r

47

n
(2.3) n’z/r 2 {I UrxzdFk(x) - (I

2
x dF (x)) ] d o.
k=l IxI<n l/r k

xI<n
Remark 2.6. Lemma 2.1 has already been used in the proof of Lemma 1.3.

Section 2.3 Main Results.
Let (Kn: n 2 I] be a sequence of independent random variables

n
and write Sn = kflxk. Take 0 < r < 2. Set qn(x) = Pr(IXnI > x),

q(x) = Pr(IXI > x), and Fn(k) = Pr(Xn S x), F(x) = Pr(X S x). Let
L(') be a non-negative, non-decreasing and continuous function of slow
variation. In this section we state the main results of this Chapter
and indicate their relationship with the results obtained earlier. The

proofs of these theorems will be given in the next section.

 

Theorem 2.3.
(a) Let qn S q .and for t 2 0 let nt+1L(n)Pr(IXI > nl/re) ~ 0
a__ n ~" for all e > o and (2.2) and (2.3) hold. Then
ntL(n)Pr(ISnI > nI/re) * o 23_ n w a for all e > o.
(b) Let qn s q 5351 for t' 2 o .13 EEIXIr(t+1)L(IXI)] < a
O
and (2.2) and (2.3) hold. Then 2 nt-1L(n)Pr(ISnI > nl/re) < G
n=1
for all e > 0.

Theorem 2.4.

t l/r
(a) For t 2 0, let n L(n)Pr(ISnI > n e) 4 o as_ n w a
n
for all e > 0. Then ntL(n) Z Pr(IXkI > nl/rc) * o for a l
- k=l

E > o and (2.2) and (2.3) also hold.

a
(b) For t > o, if E nt-1L(n)Pr(ISnI > nl/re) < w for al

n=1
3 > 0 then EEIXkIrtL(IXkI)] < 9 .EEE each R.

48

Theorem 2.5.

For t 2 o the following statements are equivalent:

1Me) -» o for al a > 0,

V

(a ntL(n)Pr(ISnI > n

(b) ntL(n)Pr( max IS I > nI/re) * 0 or al 3 > o.

15kSn k

‘If t > o, the above statements are_gguivalent.£g

(c) ntL(n)Pr(Sup k-1/r|sk| > 6) —o 0 for 81 e > 0.
k2n

Theorem 2.6.

For t 2 o the following statements are equivalent:

a

(a) z nt-1L(n)Pr(ISn - med(Sn)I > sure) < a £4 31 e > 0.
n=1
" 1 1/

(b) 2 nt- L(n)Pr( max ISk - med(Sk)I > n r6) < a or a 1
n=1 lSkSn

e > o.

If t > o, the above statements are equivalent to

a
(c) 2 nt‘1L(n)pr(sup k
n=1 an

-1/rISk — med(Sk)I > e) < a for al

a > o.

‘lf moreover t 2 1, then the following statements are equivalent:

1 °’ t-1 1/
(a ) 2 n L(n)Pr(ISnI > n re) < a or all e > 0.
n=1
1 °' 1: 1 1/
(b ) 2 n - L(n)Pr( max ISkI > n re) < a or al a > 0.
n=1 lSkSn
1 ° t 1 1/
(c ) E n - L(n)Pr(sup k- rISkI > e) < a for all e > 0.
n=1 an

Rema rks .

2.7’ Baum and Katz [2] have obtained the r = l, L(x) = 1 case of

49

Theorem 2.4b as well as the (a), (c) and (a1), (c1) equivalence parts

of Theorem 2.6 for the r = l, L(x) = 1 case.

2.8 In the case of independent and identically distributed random
variables Theorems 2.3a and 2.4a yield the (a), (b) equivalence part of
Theorem 1.1 and Theorem 2.3b yields the (8) implies (b) part of Theorem

1.2.

2.9 The result of Theorem 2.4b cannot be improved. This follows triv-
ially by considering the sequences for which Xk = o, k = 2,3,... and

I rt+5
1

Elelrt<~ but EIX =e for all 6>o.

2.10 The r = l, t = 0 case of Theorem 2.4b is worth further mention.

O
Baum and Katz [2] have established that 2 n-lPr(ISnI > nc) < a for
n=1

+
all e > 0 implies E log IXkI < 0 for all k. Because of techni-
calities in the proof this result is not amenable to generalization for

r # 1.

Section 2.4 Proofs.

In this section we outline the proofs of the theorems stated in
section 2.3. The methods used here are the same as already used in
Chapter I; namely a systematic use of truncation, symmetrization and
Markov inequality. ~We will therefore conserve Space by sketching the

proofs wherever they differ from the proofs of Theorems 1.1 and 1.2.

Proof of Theorem 2.3.

To each random variable Xk, k = l,2,...,n, assign a random

. 8_ ' '..
variable Xk - Xk Xk, where Xk :3 independent of Xk and has the
same distribution and write 88 = 2 Xi. Now notetfimu;the hypotheses
1

n
k:
in the case of both (a) and (b) imply n-l/rsn 2 0. Consequently, from

50

the weak symmetrization inequalities it follows that it suffices to

1/

r
e) m o for all e > o in the case of

3
show that ntL(n)Pr(ISnI > n
a
1/r

a) < w for all e > o in the case
l/r

(a) and 2 nt'1L(n)Pr(Is:| > n
“'1 t+1
of (b). Also, in the case of both (a) and (b) n L(n)Pr(IXkI > n

e) v 0
for each k and we may use Lemma 1.2. The rest of the proof is essen-

tially that given in section 1.4 for Theorems 1.1 and 1.2; only the

details are different. We omit these details.

Proof of Theorem 2.4.

To prove (a) we note that the hypothesis implies n-I/rSn E o

and hence that n-UrX.k E o for each k S n. Therefore it is sufficient

n
to show that ntL(n) Z Pr(IXiI > nl/re) w o for all e > 0. Pro-
k=l
. . . . s l/r
ceeding as in section 1.4 we write A1 = (Xi > n e) and
n
B1 = ( 2 X? 2 o) for i = 1,2,...,n and obtain
j#1,j=1 J
_ n n
Pr(SS > nL/re) 2 E Pr(A,)I;l - Z Pr(A )1.
n . i 2 J
i=1 j=l
-l/r s P n
Since n Sn ~ 0, Lemma 2.1 yields 2 Pr(AJ) fl 0. Therefore, for

i=1
5 > o arbitrarily small we can choose an N so large that for n 2 N
n

Pr(SE > nI/re) 2 6% ' 5) .2

Pr(A,).
J=1 J

This completes the proof of part (a).

In the case of (b) we have for any fixed k,
° 1
m > 2 nt- L(n)Pr(Sn 2 2n
n=1

1/

re)

Q
2 Z nt-1L(n)Pr(S 2 2n
k n

n= +1

l/re).

51

Now for all n 2 k we have

n

Pr(Sn 2 an/re) 2 Pr[(Xk 2 4n1/re) n ( 2 x. > -2n1/r€)]
J*k,J=1 J
n
= Pr(Xk 2 4n1/re)[l - Pr( 2 X, S -2n1/r€)]
j#k2j=1 J
n
2 Pr(Xk 2 4n1/re) - Pr( 2 X, S -2n1/r€),
j¥k,j=1 J
so that
a 1 1 n
(2.4) 2 nt- L(n){Pr(Xk 2 4n /re) - Pr( 2 X, S -2n1/rc)} < a.
n=k+l j¥k,j=l

Again, we have from the assumption of independence,

a n
(2.5) 2 nt'1L(n)Pr( z x. s -2n1/re). Pr(xk s nl/rc)
n=1 j#k,j=1
a
S 2 nt-1L(n)Pr(Sn S -n1/r€) < a.
n=1

Let 1 2 6 > o. For each fixed k there exists an N so large that

k
Pr(Xk S nllre) 2 6 for all n 2 Nk' It follows from (2.2) that
a n
(2.6) 2 nt-1L(n)Pr( Z X, S -2n1/r€) < N.
n=1 j#k,j=l J
a t 1
Using (2.6) in (2.4), it immediately follows that 2 n - L(n)Pr(Xk 2 4n
n=1

for each k. A similar argument shows that for each k,

° t 1
2 n - L(n)Pr(-X.k 2 4n
=1

l/re) < a.

n

Thus we have EEIXkIrtL(IXkI)] < a for each k.

1/

re) < w

52

Proof of Theorem 2.5.
A careful analysis of the (b), (c) and (b), (e) equivalence parts
of the proof of Theorem 1.1 shows that the "identically distributed"

part of the hypothesis has not been used.

Proof of Theorem 2.6.

Once again we analyse the proofs of relevant parts of Theorem 1.2
and obtain a proof of Theorem 2.6. We observe that in the (b), (c)
equivalence part of the proof of Theorem 1.2 we used the hypothesis of

n-l/r P

identically distributed summands only to show that 8n n 0.

However, this part of the hypothesis was not needed to conclude

n-1/rSz 3 0. Thus in the present case we can use n-l/rS: E 0 but
-l/r P . . .
not n S 2 0 unless t 2 1. This explains the difference be-

tween the corresponding parts of Theorems 1.2 and 2.6.
Similar remarks hold for the proof of (b), (e) equivalence part

of Theorem 1.2.

Remark 2.11. Parts of Remark 1.6 are also relevant in the context of

Theorems 2.5 and 2.6.

Section 2.5 Exponential Convergence Rates.

Let (Kn: n 2 1} be an arbitrary sequence of random variables
n
and write Sn = 2 Xk. Take 0 < r < 2. We will say that the sequence
k=l
{Kn} converges exponentially fast £2 zero if there exist constants A

and p < 1 (depending on e) agch that

1/r

l/re} S A pn , n = 1,2,...

(2.7) PrIISnI 2 n

This generalizes the notion of exponential convergence introduced by

53

Baum, Katz and Read in [3] (for r = l the two definitions coincide).
In this section we shall show that, in the case of independent
random variables, the existence of all the moment generating functions
on the same interval is necessary and that a growth restriction on the
product of the first n of these generating functions is necessary and
sufficient, to ensure exponential convergence. More precisely, we shall

prove the following results.

Lemma 2.2.

.LEE. {Xk] .93 3 sequence 2E independent random variables such
“that (2.7) holds for some constants A, e > o ‘gnd. p < 1,
Then, each Xk ‘hgg g moment generating function and these
moment generating functions all exist 23 some common interval

about zero.

 

Theorem 2.7.

Let [an n 2 1} .23 g sequence 2f independent random variables
n

and write 8 = Z'Xk. Then, the sgquence {X } satisfies (2.7)

___.______ n k=l n ._________.

if and only if for all e > 0 there exists a constant Me and

t6 such that

k n

n tX tS ItI nl/r
(2.8) IIEe =Ee sues e

f t -t t .

or E E 6’ 6]

Remark 2.12. The r = 1 case of Lemma 2.2 and Theorem 2.6 has been
obtained by Baum, Katz and Read [3].

Proof of Lemma 2.2.

The proof of Lemma 2.2 is essentially the same as that of Theorem

54

2.4b and is a modification of the one given by Baum, Katz and Read

[3, 188]. Proceeding as in the proof of Theorem 2.4b, we note that

l/r n

(2.9) An“ 2 Pr( 2 xk 2 2n1/‘e>
k=l
I1
2 Pr(Xk 2 4n1/re) - Pr( 2 X. S -2n1/r€)
j¢k.j=1 J

for all n 2 k. From the assumption of independence, it follows that

n
(2.10) Pr( 2 'X S -2n
J#k,1=1 j

l/r

l/r e) s Apn

l/r l/r

e)Pr(Xk S n

e) S Pr(Sn S -n

Let 1 > 6 > o. For each k there exists an integer Nk so large that

Pr(Xk S nllre) 2 5 for all n 2 Nk’ and it follows frcm (2 9) and
(2.10) that for n 2 Nk’
l/r l/r
(2.11) Pr(Xk 2 4n1/re) s Ap“ + A6 1p“
l/r
s 13pn

For t > 0, we have

+
t
seXkS1+Izetderuka>

a 4'1/1: tx
1 + XII “1 el/r e dPr(Xk S x)
3‘1 4(1-1) s

l/r

4tj

a
S l + 2 e ePr(Xk 2 4(j-1)1/r€)

j l
Nk
S l + E

j:

l/r 0 .1/r
e4teJ + B 2 (e p)J
l j=l

55

t:
if e4t€p < l. A similar argument shows that E e < a, and it

th

follows that E e < a for all t e [-to,to] with to determined

independently of k.

Proof of Theorem 2.7.

We again use the methods of [3] to obtain a proof of Theorem
2.7. Given 6 > 0 take 0 < 61< e (to be chosen later) and for
t > 0 write

ts+
n a tx
S S
E e 1 +-Io e dPr(Sn x)

a l/r
= 1-+ 2.Ik 51/r etder(Sn s x)
k=l (k-l) 5
a l/r
s 1 + z et(k+1) 5Pr(s 2 kl/ré)
n
k=0
0 (j+1)n-l l/r
= 1 + 2 2 et<k+1> 5Pr(s 2 kl/ré)
j=o k=jn n
9 l/r , l/r
s 1 + z n etén (3+2) Pr(S 2 (jn)1/r6)
j=o “
l/r “ . l/r l/r
(2.12) S 1 + n[et6n + jgoet(J+2) n 6Pr(S 2 nl/r(j+l)I/r6:
' n
To estimate Pr(Sn 2 nl/r(j+l)1/r6) we first note that

l/r l l/r

6) S Pr(Sn 2 E-n. ,l/r

Pr(Sn 2 nl/r(j+l) (J +%)5)

l/r

(2.13) s pr(sjn 2'%(jn) 5)

l/r

_1_ .l/r
+ Pr(Sjn - Sn 5 -4 n (J +1)6).

This follows since

 

56

Pr(sn s é-n / (j 1/r+—)5)

l/rm l_. l/r ‘l l/r
2 Pr(Sjn S %(jn ) -(Sjn - Sn) S 4(Jn) 5 + 4 n 6)

l/r l/r

2 Pr(Sjn S'%(jn) 5)Pr(S - Sn 2 --%(j1/r+l)n ),

jn
so that

l/r /r6) s 1 - Pr(sn S-ln ”/ (j 1/

r
2 +—>6>

Pr(Sn 2 n (j+l)1

s 1 - [1 - Pr(S 2'%(jn)1/r5)][1 - Pr(S l/r+1)n1/r)]

5
jn - 8n S -‘Z(j

jn

l/r l/r

s Pr(sjn 2-%(jn) 5) + Pr(Sjn - sn s --%5(j1/r+l)n ).

Again, by independence

l/r1 nl/r r16l/r

1
Pr(Sjn - Sn S - 26(j ). Pr (Sn s—

2» Pr(S s - ion)”

in 6”

and since

P (sn s— 1”5) 2 1 A “l/r
r n ' 6/495/4

>
Z n > °

for sufficiently large n, it follows that for large n and all j = 1,2,...,

l/r(jl/r+1)) S (nj)u Mn

1
(2.14) Pr(S - Sn S -‘Z5 n 2516/49,”4

jn

Using (2.7) and (2.14) in (2.13), we obtain

l/r (nj)u

. l/r _ p(nj)
(2.15) Pr(Sn 2 n (J+1) 5) S A5/495/4 r(1+n1) _ 5p5/4

for large n and all j. Using this estimate in (2.12) we have

57

+
tS l/r
(2.16) E e n s 1 + n[et5n + Ba 2 et(j+2)
1‘0

1/rnl/r6 (nj)l/r]

p6/4

for large n and t > o. If 0 < r S 1 then we have for large n,

 

 

ts+
E e n
l/r l/r 2/r-1 9 l/r-l,l/r l/r l/r
s l + n[et6n + Bsetn 2 5 2 e5t 2 J n . pgyz) J
3‘0
l/r l/r 2/r-l
(2.17) s n[2etén + B etén 2 .2. 1 J,
6 . l/r-l l/r
t52 n
1-(8 pé/a)
6121“"1
where we have chosen t > 0 such that e 95/4 < 1. 0n the
other hand, if 1 S r < 2 then for large n we have
ts+
E e n
l/r l/r l/r “ l/r .l/r
s 1 + n[et5n + B et2 n 5 2 (et5n p )3 ]
5 5/4
1‘0
t5 l/r
l/r 1/r 1+(e p )
(2.18) S nEZetén + Béet6(2n) t6 6/4 Ilr 2 J,
n

where we have chosen t > 0 such that etépél4 < 1. From (2.17) and
(2.18) it is clear that for o < r < 2 there exists a constant D5’
say, such that
+
tS 2/r-l l/r
Ee nSnD6e2 ItIn 5
for t in some interval about zero. A similar argument shows that
tS- , 2/r-l l/r
Ee “5:11)an ItIn 5
for t in some interval about zero. It therefore follows that there

exist Me and t6 > 0 such that

58

S 1
E et n S M eltlen /r
6

for t 6 [-t€,t€] and the necessity is proved.

For the sufficiency, we use the well known Markov type inequality

[23, 158]
l/r t(Sunnl/re)
Pr(Sn 2 n e) S E e
fot t > 0. Choosing 5 < e we obtain
Pr(Sn 2 nI/re) S exp(-tée nl/r).M6.eXp(5t5n1/r)

nl/r
= M6Eexp t6(6'€)]

1/r

A similar argument shows that Pr(Sn S -n e) converges to zero

exponentially fast and the proof is now complete.

Section 2.6 Miscellany.

Let X1,X2,... be independent random variables and write
n
S = Z , M = max 8 , and N = max IS I. In this section, we
“ k=lxk “ lSkSn k “ lSkSn k

give some simple results concerning the convergence of Sn’Mn and NH

in probability. In view of Remark 2.11 it follows that

x N E o if and only if x-1 S E o

n n n n

where xn is a sequence of positive real numbers with xn d 0 as

n w 9. Also, we trivially have

x'lN go implies x'lM 3
n n n n

0.

These remarks lead to a new proof of the following theorem due to

Heyde [14].

59

Theorem 2.8. (Heyde)

Let {Kn} '23 a sequence 2f independent random variables Wlth

 

finite expectations un = E Xn such that. n-1(u1+u2+...+un).w u

_g§- n w m, where u 2_o .lfi finite. _If n-ISn E u .gg n m “,
then n-IMn E u and nilN “ u. ' ”

. n .

.ELQQf; Let n-ISn E u. write Y1 = Xi - u and set
n
2 = Z‘Y, = S - nu. Then, n-IZ g o, and by the remarks preceding
n i=1 i n n
the theorem it follows that n-1( max IZnI) E o. It therefore suffices
lSkSn

to show that

Nn - nu S max IZkI

lSkSn
and
M - nu S max I2 I.
“ lSkSn k
We have,
Nn - nu S max {ISkI - ku}
lSkSn
S max IzkI:
lSkSn
since

ISkI S ku + S - ku .

k
Again, since

-IskI s Isk - ku - ku,

we have

-(Nn - nu)

60

max IS

lSkSn

nu kl

nu + min {-ISkI]
lSkSn

IA

min IISk - kuI + (n-k)n}
lSkSn

IA

min IZkI

lSkSn
S max IZkI.
lSkSn

To complete the proof we show that

Mn - nu S max IZ

First, note that

- S
Mn nu Nn

Also,

-(Mn - nu)

and the proof of the theorem is
Next, if we write

Tn = max I

lSkSn

and

I.
lSkSn k

- nu S max I2

I.
lSkSn k

up +' min ('3
lSkSn

k)

V\

min (Isk - kuI + (n-k)u)
lSkSn

min I2 I,
lSkSn

IA

complete.

S min ISkI

I -
k lSkSn

61

R = max Sk - min Sk
lSkSn lSkSn

then we are led to the following extension of some earlier results

(see Remark 1.6).
‘Lgt {Kn} ‘QE 2 sequence 2f indgpendent random variables and
[xn], {an} ‘bg sequences pf positive £531 numbers 3125 an 2 c > o
22221.1. n 224 x,” a.s. n-w- flea.

> fl > ‘
(a) anPr(ISnI xne) o for all e 0 implies

' > —o > .
1) anPr(Rn xne) o for all e o, and
ii) a Pr(T > x e) H o for all e > o.
n n n ----
G
> > v o
(b) nElanPrdSnI xne) < a for all e o .lmplies
m
i) zaPr(R >xe)<eu forall e>o; and
n n n ---- -—-
n=1
a
ii) 2 a Pr(T > x e) < a for all e > 0.
n=1 n n n ----

Finally, let {KR}, {XL} be two sequences of independent random

variables such that for every k, Xk and XL are identically distributed.
n n

Let Sn - kflxk, 8; = kEIXL, 2n = 8n + 8; and Yn = Sn - 8;. In [7]
Egbek compares the rate of convergence of {Yn} and {Zn} to their
correSponding limit normal variables under the assumption that each of
the random variables Xk has moments of all orders and satisfies
Lyapounov's condition for third central moment. It is of interest to
compare the rate of convergence of the sequences {Yn} and [Zn] in

the law of large numbers. By methods used above we easily obtain the

following result without any moment conditions on the Xk's.

Let {xn], {an} ‘29 sequences‘gf positive real numbers such that

62

an 2 c > o for all n and xn * m ‘§§_ n m w. Then

(i) anPr(IZnI > xne) * o for all e > 0 implies

> -* >
anPr(IYnI xne) 0 or al 6 o

and conversel ,

Y > _. > o o
anPr(I nI xne) o for al e 0 implies

- > —. >
anPr(IZn 2 med(Sn)I xne) o for al 9 o.

a
(ii) 2 anPr(IZnI > xne) < a

for all e > 0 implies
n=1
a»
>
nElanPrdYnI xne) < G for all e > o

and conversely

H

2 a Pr(IY I > x e) < a for a
n n n n -—-

l e > 0 implies

i anPr(IZn - 2 med(Sn)I > xne) < a .£2£.ill e > o.

In particular if all the random variables Xk are symmetric,
then the sequence {Yn} converges at the same rate as {Zn}. However,
if at least one random variable Xk is not symmetric, then the sequence
{Yn} converges more quickly than the sequence {Zn}. These prOperties

were noted by Ugbek in her more restricted context.

CHAPTER III
CONVERGENCE RATES IN THE LAW OF LARGE NUMBERS FOR

WEIGHTED SUMS OF INDEPENDENT RANDOM VARIABLES

Section 3.1 Introduction.
Let IX“: n 2 1] be a sequence of independent, but not nec-

essarily identically distributed, random variables and let {an k: n,kzl}

a J
be a double sequence of real numbers. Write S = 2 a XR (and note
- k=l n,k

the change in notation). In [10] Franck and Hanson study, under fairly

mild conditions, the rate of convergence of the sequence {Pr(ISnI > 6)}
to zero for the case t > 1 (t is a constant to be defined in section

3.2) and remark "For completeness, the case t S 1 should be thoroughly
treated for the coefficient sequence an,k'°'"' The present work is in

the Spirit of Theorems 1 to 5 of [10] and is intended to complete these

theorems for the case 0 < t S 1.

In section 3.2 we introduce the notation and state some lemmas
which are needed in the sequel. In section 3.3 the main theorems are
stated; these are proved in the next section. The relationships be—
tween the theorems stated in section 3.3 and the results obtained

earlier are examined in section 3.5. The final section is devoted to

some miscellaneous related results.

Section 3.2 Preliminaries.

Let Xn,a and Sn be as defined in section 3.1. Suppose

n,k’

that C,a,B,p and t are constants such that

(3.1) o<tSl,B>o,p>o.
a

(3.2) 2 Ian k| s c n“.
k=l ’

63

64

-B
(3 3) SEP Ian’kI s c n .

a
(3.4) 2 la It s c n‘p.

k=l “’k

We recall that C denotes various positive constants whose exact

values do not matter. Note that for o < t S l

[Sip lamkltl s 112:. lawnt 2 If Ianjkl“

so that we may assume

(3.5) B 2 p/t and B 2 -a.
»Also, since

EIan)kI = 123 flatbkll"t IaMI‘}
213:.) Ian)k|]1'tilan)k|t,

we observe that

(3.6) (p + oz) + 8(1-t) S 0.

Note that since o‘< t S l, (3.6) implies
(3.7) p +-a S 0.

Unfortunately, due to technicalities of the proof we have to

assume further that there exists a constant A, o < A < t such that

(3.8) 2 I IA
k

S .
an,k C

This is quite a plausible assumption in view of (3.4).

65

For notational convenience we write

F(y) = SUP Pr(IXkI 2 y).
k

We have the following simple lemmas which we state without proof.

Lemma 3.1.
Let 0 < t S l and th(y) S M.< O for all y > 0. Then
EIXka for each A, o < A1< t:.l§ uniformly bounded 13 n

and k.

Lemma 3.2.
Let 0‘< t S 1 and th(y) S M.< O for all y > o. For any
5 > 0 define

-l
Ian,kl

if S n
Y . {xk .__.. kal
n,k 0

otherwise.

Then EIYn ka for each fixed I, o < I < t, is uniformly
)

bounded.ip n and k.

Remark 3.1. Throughout this Chapter we shall assume that only those

k's are considered for which an k ¥ 0. -According to this convention,
)

summations will be taken only over those values of k for which

a f o and the definition of Y in Lemma 3.2 makes sense.
n,k n,k

~Section 3.3 Main Results.
Theorem 3.1.
.If p > o, B > o, o < t S l and th(y) S Mi< a for all

y > 0, then for every 6 > o,

(3.9) Pr(ISnI > e) s 0(n’p).

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

66

Theorem 3.2.
Let p > o, B > o, 0‘< t S l and th(y) ” 0 pg_ y w’“. Then,

for every e > 0,

II
0
A
:3

Pr(ISnI > e)

Theorem 3.3.

 

 

Let p > o, B > o, o < t S l, p + a < o and suppose that F
satisfies

lim F(y) = o {223. I: ytIdF(y)I < '.
y—a

Then for every 9 > o,

 

.
z np‘lrr(|s I > e) < a.
n=1 n

Theorem 3.4.

 

Let p > o, B > o, o < t S 1, and suppose that F satisfies

 

11m F(y) = o aid I: yldF<y>| < 2.

‘_£ there exists £_non-negative and non-decreasing real valued

function G satisfying (3.11) and such that C(x) 2 F(x) and

x21 y2x xtF(x)
where Y ‘lg a constant, then (3.12) holds for every 52> o.

 

 

Theorem 3.5.

.If p > o, B > o, o < t S l, and F satisfies

lim F(y) = o and I: yt10g+yIdF(y)I < a,
H

67

then (3.12) holds for every e > 0.

Remarks.
3.2 In Theorem 3.4, the function F automatically satisfies (3.11)

since 0 < t S 1.

.gpg We wish to draw attention to the formal similarity of these theorems
to those proved in [10]. In [10], Franck and Hanson prove Theorems 3.4
and 3.5 for the case t = 1 also. For this particular case they use

the stronger conditions of these theorems to yield (3.12) without using

condition (3.8).

Section 3.4 Proofs.
The method of proof used here parallels that of Franck and Hanson
[10]. -We repeat, in order to emphasize the fact, that the summations

are taken only over those values of k for which an k ¥ 0. Our first
3’

step is to observe that with Yn defined in Lemma 3.2, and in the

,k

notation of section 3.2, we have

Pr(ISnI‘2‘2e)

(3.15) s 1Errdan’kxkl > a)
(3 16) + 2 Pr(Ia I > n-b) Pr(Ia .x | > n-6)
- , :1,ka ° m“ j '
J'r‘k
(3.17) + Pr(IE amkymkI > a).

We omit the proof of this since it is essentially the same as the proof
of inequality (1.7).
As in [10], the next step is to examine each of the expressions

(3.15), (3.16), and (3.17) separately and show that the rate af which

68
each approaches zero is appropriate for the theorem in question.
.Expression 4(3.l§),

For Theorems 3.1 and 3.2, we note that the expression (3.15) is

bounded by
2—2— [-———- Pr<| |> T—‘Ie )3
k Iae It x“ an,k

S C 2 Ian kIt sup IEytPr(IXkI > Y)]
2 ychan’kI'

-p
S C n [sup sup {y tPr(I I > Y1]-
k y2CnB xk

The assumptions of Theorem 3.1 guarantee that the quantity in the square
brackets in the last expression is bounded and it follows that (3.15)

is 0(n-p). ~Similarly, the assumptions of Theorem 3.2 ensure that this

same quantity in square brackets is 0(1) and so (3.15) is o(n-p).
As for Theorem 3.3, we see that
(3-18) 2 n - 2 Pr(Ia I > 3)
n=1 k=l “’kxk
S 2 up.1 2 F(eIa kI-l)
k n)
n
s ZF(m-l) 2 hp‘1
m [(n,k):(m-l)<eIa I'lSmI
n,k
9-1
(3.19) = z (F(m-l) - F(m)} z n .
m [(n, k): eIan kI'ISmI

Since am"1 5 Ian kI 5 C n'B, the set {(n,k): eIa '1
)

s O

n,k| m] W111 be
1/8 . -1

non-empty only if n S [C m/e] . Furthermore, Since a m S Ian kI
)

69

and 2 Ian kI S C no, it is clear that for fixed values of n and m
k

)
there are at most C m 60/6 pairs (n,k) over which the second sum-
mation in (3.19) is taken. It follows that (3.19), and therefore (3.18),

is bounded by

[Cm/€11/B a

 

2 {Fun-1) - mm) 2 C “ m mm
m n=1 6
[C m/CJI/B
= C 2 mt{F(m-l)-F(m)}[m1-t 2 na+p-l]
m n=1

c 2 mt{F(m-1) - F(m)}
m

IA

where we have used the assumptions that p +~a < o and (p+a) + 8(l-t) S o

to show that

[C m/SJI/B
ml-t 2 nU+P-1 = 0(1).

n=1

For Theorem 3.4, we use (3.13) and find that (3.18) is bounded by

9-1 .
2 B}n E Pr(Ian,kaI > e)

 

In:e<Cn-
9-1

+ Z n 2 Pr(Ia I > e)

[n:eZCn-B} k n,kxk
(3.20) S 2 -B np-l E'Pr(Ia kku > e)
[n:e<Cn I k n,
B t B

+ 2 'B np-l 2 (an IQ) C(arliéC)y

[n:€2Cn I k (clan,k| )

Since B > o, the set In: a > C n-B} contains only a finite number of

elements, and the first term in (3.20) is then finite if

70

ZPr a > . . O O
(O .
k (I n’kaI e) for each n That this indeed is the case is

seen as follows: For each n,

WM

Pr(Ian,kaI > e)

S )3 F(s:Ian kIul)
R

J

S EN!!!) 2 _1 1
m {k:m<eIa I Sm+1]
n,k

2 [F(m)-F(m+l)] z 1

(3.21) .
. -l
m [k.cIan’kI Sm+1)

0’
Since 2 Ian kI S C n it follows that the number of elements in
k 3

(k: eIan kI-1 S m+l] is at most C na(m+1)/e. Thus, (3.21) is bounded
)

by

(0 na/e) 2: (m+1){F(m) - Form}
In

S C nu I: yIdF(y)I.~

which is finite for each n.

As for the second term in (3.20), we have

“9-1 2 (gas/cfcmnem

 

Y _ -
In=62Cn B} k (eIan’kI 1)t
S C 2 B np-I+BEG(enB/C) 2 Ia kIt
(n:en 2C} k n,

S C 2 np-1+BtG(enB/C). n.-p
n

s c z nBt[G(enB/C) - G(e(n+1)B/C)]

n
s c I: xtIdG(x)I

< a,

 

71

In case of Theorem 3.5, we return to (3.19) and note, as before,

that there can be terms in the second summation in (3.19) only if

1/8

-1 t t t ‘P
n S [C m/e] —Also, (a m ) S lan,k| and Z lan,k| S C n to-

k
gether imply that there are at most (C n pmt/et) elements in the set

[(nyk): eIan kIm1 S m] for fixed n and m. Thus (3.19), and there-
1
fore (3.18), is bounded by

[Cm/6]”B t
c 2 [F(m-1) - rum} 2 m n

m n=1

-1

S C 2 mt[F(m-l) - F(m)] log m
m

S C I: xt log+xIdF(x)I

<0.

Exppession (3.16).

We note that the expression (3.16) is bounded by
t 5t -6 -1 t -6 -1 2
(3.22) [E Ian,k| n [(n Ian,k| ) Pr(IXkI > n lan,k| )]] .

The hypotheses of each of the five theorems guarantee that th(y) S M‘< O
for all y > o, and this ensures a bound on the quantity in the square

brackets in (3.22). Thus (3.22), and therefore (3.16), is bounded by
(3.23) cIE Ian,kItn6t}2,

and hence by

(3.24) c n2<6t‘p).

If we choose 5 such that

 

 

72

(3.25) o.< 6 < (p/Zt)

in the case of all five theorems, we see that (3.16) is o(n'p) in
the case of Theorems 3.1 and 3.2, and
9-1 > -6 > -5
E n jiLPrCIan’RXRI n }Pr{|an’jxj| n }

S C 2 n26t-p-1

n

<.,

in the case of Theorems 3.3, 3.4 and 3.5.

.Expression (3.lZ).
In this case we use Markov's inequality and see that for a

positive integer A,

(3.26) Pr(IE an,kYn,kl > e}
s C EEZ Ia IIY IJA
k n,k n,k

* ** a a wk
5 C 2'2 (§”k21mk) k21|an,¢(k)'mkE|Yn.¢(k)| ’

where the first summation is taken over all integers a,m1,...,m,a Such

that mk2 l, k = l,2,...a, Z‘mk = A, and distinctsets of integers
k
[m1,...,ma} appear only once in the sum. The second sum is over all

1 — l mappings m from {l,2,...,a} into positive integers. Recall
Ix

that Lemma 3.2 ensures a uniform bound in n and k on EIYn k
)

for each fixed K, o < K < t. Choose k to satisfy (3.8). -We have,

“k ”k
‘3'“) Iawml E'Yn,cp<k>'

-K
S C Ian:¢(k)|x[ sup Ian2k||Yn:k(w)|]mk

w,n,k

73

S C la n ,

n,¢(k)|

where C is a constant which depends on A and A but not on n and

k. Thus, (3.26) is bounded by

a X -5( -X)
C 2*:** H la :n mk
k_1 n,¢(k)

a
2*n-6 (A-Xa) 2** 1'1 I a

k=l n,CP(k)

S C

where we have used the fact taat the multinomial coefficients

a
(All H mk) are bounded by a constant depending on A but not on n
k=l
or the a '8. Now, we note that
n,k
** a A A a
s s c
2 kgllan,¢(k)| [If lan,k| J )

and it follows that (3.26) is bounded by

(3.28) c 2*n'5(A"‘a).

If we now choose A so large that for T > p,

A > T 6'1(1->.)'1,(o < 6 < p/2t),

* -T
tlen 6(A-Ka) > T and each term in the sum 2 will then be o(n ).

*
Since 2 has only a finite number of terms, the number depending on

A but not on n, the proof of all five theorems is complete.

Section 3.5 Some Remarks on the Relationships Between These Results
and Results Obtained in Chapter I.

Theorems 3.2 and 3.3 may be Specialized to yield some results
obtained previously. Let 0CD: n 2 1} be a sequence of independent,

identically distriubted random varialbes and take 0 < r < 1. For

74
= n-l/r for k S n and a = o for k > n
n,k n,k
and take a - l - l/r, B - l/r, p = t, then, for o < t S l/r - l
nl/r

example, if we set 8

Theorem 3.2 yields the result that nt+1Pr(IXI >
l/r

) m 0 implies

ntPr(|SnI > n e) d o for all e > o. This is a part of Theorem 1.1.

Again, for independent and identically distributed random variables

X“, if we set a = l - l/r, B = l/r, p = t and take 0‘< r < l, o < t S l/r - l

D
in Theorem 3.3 then EIXIr<t+1> < 0 implies 2 nt-lPr(lSn| > nI/re) < I
n=1

for c > o. This is a part of Theorem 1.2.

Section 3.6 Miscellany.
Let -X1,X2,... be a sequence of independent random variables and

[a : lSkSn,nzl} be a sequence of real numbers such that max 8 m o
n,k n,k
n lSkSn
as n m C. -Write S = 72 a Xk, and note the change in notation.
k=l n,k
Also, note that an k's are not assumed to satisfy the conditions of
. )

section 3.2. .We remark that the condition maxlan kl
k I

to the condition sup an R," 0 which has been shown to be a necessary
k ’ '
and sufficient condition that 2 a Xk’g o in the Special case when
k=l n,k
is a Toeplitz summation matrix [25].

m o is similar

< a
Elxkl O and A (an,k)
In this section we give some results which go in the converse

direction to the results of section 3.3. More precisely, we establish

the following theorem.

Theorem 3.6.

For 9 2 o,

(a) inr(ISnI > e) w 0 for al a > 0 implies

n .
P
2 P > ~ 11 > ,
n k=l r(lan,kxk| e) 0 or a e o

a
P > . .
(b) 2 n Pr(ISnl e) < a for al 6 > 0 implies

n=1

75

a n
2 np 2 Pr(la I > e) < 0.
n=1 k-l “’kxk

We defer the proof of Theorem 3.6 until some lemmas have been

established.

Lemma 3.3.

P n
‘3; sn a 0, then kflrr(|an:kxk| > c) S 0.

Proof: Let us write X = a Xk. Since the random variables
--- n,k n,k

Xn k are also independent, it follows by the degenerate convergence
J

criterion [23, 317] that it will be sufficient to show that Xn k
)

satisfy the u.a.n. condition (see [23, 290] for the definition of u.a.n.).
This will certainly be the case if max Pr(IX.n kl > e) w o for every

k )
e > o [23, 302]. Now, for each i S n,

n
Pr(sn > 2;) 2 Pr[¢.n,ixi > 4e) n (j¢i?j=13“’jx5 > -2e)}

n
= Pr(a .X. > 4e)[1 - Pr( 2 a X 5 '26)]
n,i 1 j¢i,j=l n,j j

n
2 Pr(a X, > As) - Pr( 2 a -X S -2:).

It is therefore sufficient to show that

n
Pr( 2 a X S ~29) ~ 0 as n q’fl

j¥iJJ=1 n’j j

for each i. -We have

n
Pr(-”Hi:j=1 an’ij S -2c)Pr(an,iXi S e)

S Pr(Sn S -e),

and since a , m o as n w G, Pr(a X. S e) m l as n ~ ~.
n,i n,i 1

a .u 1.! {t l 4" .11. i A!“ ‘II‘ .‘ .II' in It Iii

 

76

n
Finally, since Pr(S S -e) w o, it follows that Pr( 2 a X S -25) ~ 0
n j*i,j=1 “)3 j

and the proof of the lemma is complete.

Lemma 3.4.

Sn 3 0 implies
n
Pr(ISnI > e) 2 T] 2 Pr(Ian kxkl > e)
k=l ’

where n > o ‘;g some constant.

 

Proof: Using the weak symmetrization inequalities we see that

n
P s s s
n o where Sn 2 an,kxk’ Xk being symmetrized

k=l
n
S 8 .
Xi > e} and Bi { .2 an,ij 2 0}. We obtain
j*1:j'1

Sn 2 0 implies S

Xk. Set A = {a

i n,i

as in (1.4),

n i-l
Pr(Ss >'e) 2 2 Pr(A )[Pr(B.) - Z Pr(A )]
n i i j
i=1 j=l
n 1 n
2 2: Pr(Ai)[§ - 2 Pr(Ajn.
i=1 j=l
n
An application of Lemma 3.3 now yields 2 Pr(Aj) w o as n ~ '
i=1
and it follows that for large n,
s n s
> >
Pr(Sn e) 2 n i:1Pr(an)ixi e),

where n > o- is a constant. The proof is now completed with the help
of the weak symmetrization inequalities, noting the fact that as n ~ a,

med (8 ) ~ 0 and med (a ,X,) m o for each i.
n n,i 1

Proof of Theorem 3.6.

Theorem 3.6 follows immediately from Lemmas 3.3 and 3.4.

77

Remarks.
-l/r , .
3.4 If we put 9 = t and an k = n in Theorem 3.6a we obtain
2
part of Theorem 2.3a.
3.5 Let X1,X2,... be independent and identically distributed. If

-l/r
= n
n,k

implies (a) part of Theorem 1.1 and the t 2 1 case of (b) implies

we put p = t and a in Theorem 3.6 we obtain the (b)

(a) part of Theorem 1.2.

CHAPTER IV

-A LARGE DEVIATION PROBLEM

Section 4.1 Introduction.

Let [sz k 2 1} be a sequence of independent and identically
distributed random variables with law' iKX) and write Sn = kglxk.
Unless otherwise Specified, let xn(n - 1,2,...) denote a sequence of
positive real numbers with xn * a as n w a. Let X belong to some
domain of attraction [9, 168] with E.X = o if EIXI < a and write

n = Bglsn for normed Sums. In the literature (see for example [22],
and the review paper of Sethuraman [26]) the probability Pr(Ian > xn)
or either of its one-sided components is known as a large deviation
probability. An extension of this concept has recently been introduced
by Heyde [16]. .For any sequence {Xk} of independent and identically
distributed random variables and any monotone sequence {xn} Such that
x;18n E o, Heyde calls the probability Pr(lSnI > xn) or either of its
component tail probabilities a large deviation probability. Using
techniques essentially similar to the one used in the previous Chapters
Heyde [15], [l6], [17] investigated the asymptotic behaviour of the
large deviation probabilities for various types X which are not

attracted to the normal law. In particular, Heyde [17] showed that

for monotone sequences [xn}
> ~ >
(4.1) Pr(ISnl ann) n Pr(IXI ann)

as n m G, for any »X belonging to the domain of attraction of a non-

normal Stable law. In this Chapter we will investigate the asymptotic

78

79

behaviour of Pr(ISnI > ann) for certain types X belonging to the
domain of attraction of the normal law.

Let X belong to the domain of normal attraction and suppose
that E X = o and var(X) = 02. It is known (see Cramér [6]) that if

Efexp t X] <‘9 for t in some open neighbourhood of O, and if

xn = oQ/n), xn > 1, then

H

1
A(n zxn)]J: exp(-t2 /2)dt

n

3

:3
(4.2) Pr(sn 2 o f; xn) .. c exp[n .xn

as n w G, where 1(2) is a power series in z involving the cumulants

1/

of X and convergent for small 2. In fact, if xn = o(n 6) we have,

1 2
Pr(S 2 O./n x ) ~-——- exp(-t /2)dt.
“ n mjxn

See [9, 517-520]. It is clear therefore that we cannot expect
Pr(ISnI> ann) to behave as in (4.1) unless we restrict X suitably.
We will show, however, that if the tails of the distribution of X
satisfy a certain growth restriction the precise asymptotic behaviour
of Pr(ISnl > ann) is governed by (4.1). In this connection, it is
of interest to recall the work of Linnik [22]. .For sequences of in-
dependent random variables {Xn} with a common continuous probability

density g(') such that for x 2 l

9 9+1 4Q+5 1
> = = — — .
(4.3) Pr(X x) ng(u)du xp + xpll +...+ x4p|5 + O(xgp 5 e),

where p 2 3 is an integer and Aj's are constants, Linnik [22]

showed that for x 2 l, as n d a, we have

(4.4) Pr(Sn > O nx) ~r-lh'f:exp(-u2/2)du + R(x,/;),

f2?

80

where o = var(X) and R(x”/n) is a rational function of both variables

determined by coefficients Ap’Ap+1"°"A4p+5° Moreover, for
3 2

x 2 n I +1lplog n we have

(4.5) Rog/5) .. n Pr(X > o X f5) = nj‘“ fg(u)d'u.

an

In order to see why this type of growth restriction is needed it
is instructive to analyse Heyde'sfproof of (4.1) and see where it breaks

down. It is seen that the result

 

Pr(ISn |>xn 13112)
(4'6) $32 nPr(TX1>x: B :)2
11"“
holds whether X belongs to the domain of attraction of the normal law
or not. However, to obtain

Pr(ISn |>xn B)
nrrqx|>x:13)

 

(4.7) lim

use was made of the fact that in the case of attraction to a non-
normal stable law we have

(4.8) fl I xzdF(x) s c u2Pr(|X| > u),
x <u

where C > o is a constant. In the case of attraction to the normal

law, we have however (see [11, 172])

(4.9) 1imu2P1‘(LXJZ U)

uwm f x 2dF(x) —
lxl<u

so the same arguments do not apply. If we restrict ourselves to the

C888

81

(4.10) Pr(IXI > x) = x-pL(x) as x + m,
where p 2 2 and L(°) is a non—negative function

of slow variation,

which is strictly analogous to the behaviour for random variables be—
longing to the domain of attraction of a non-normal stable law, it is
possible to get around this difficulty.

In section 4.2 we will show that if {Xk} satisfy (4.10) then
X belongs to the domain of normal attraction (Lemma 4.1) and the
asymptotic behaviour of Pr(ISnI > ann) is given by (4.1) if a mild
restriction is placed on xn (Theorem 4.1). In section 4.3 we will

state some recent results of Heyde on iterated logarithm type behaviour.

Section 4.2 Results.

We first prove some preliminary lemmas.

Lemma 4.1.

.EEE ({Xk}‘bg”§ seguence 2f independent random variables Kipp”;
distribution .g_ and satisfying (4.10). .Thgn, ghg random
variables belongngg Egg domain 9: attraction g: £h§_norma1 lag,

Proof: If p > 2, the variance of X is finite and it is well

2

known that the result of the lemma holds. Next suppose that p
and note that it suffices to prove that U(x) 8 fix uzdF(u) is a
function of slow variation (see Theorem 13, p. 303 of [9]). If var(X)
is finite, there is nothing to prove so we may restrict ourselves to

the case var(X) = m. We have, on integration by parts,

82

U(x) f: uZdPr(IX| > u)

-x2 Pr(lxl > x) + 2 [:11 Pr(lxl > u)du

and it suffices to show that I: u Pr(IXI > u)du is a function of slow
variation (see [9, 273]). That this indeed is the case follows on in-

tegration by parts using the Lemma on page 272 of [9].

Lemma 4.2.

‘If Pr(IXI > x) = x-pL(x) 2E. x ~ a, then for a > o,

 

0(1) _1_f_
_£.

(4.11) f |x|°’dr(x) = 0(M(y 1%))
x <yan n a-p
0((yan) L(yanD if 01 > p,

where M(y) = Ii x-1L(x)dx is‘g function 2: slow variation.

Proof: For a S p, the proof is similar to the one used for
Lemma 1.2 and for a > p, the result follows from Theorem 1, p. 273

of [9].

Lemma 4.3.

3:, Eg_ n m a, n Pr(le > xn) d 0, then
(4.12) Pr( max ~|xk| > xn) .. n Pr(|x| > xn).

lSkSn

Proof: Since

Pr( max I I > x ) = 1 - Pr(max l | S x )
lskSnxk n k xk n
n
= l - Pr H (I I S x )
k=l xk n
= 1 -

I1
11 Pr(l 1 s x)
1:1 ‘4 n

83

n
= 1-II[1 - Pr(l | > X )3:
k=l Xk n

the assertion follows by passing to the limit in the elementary inequality

We are

(4.13)

(4.14)

n n
1 - exp{- 2 Pr(] | > x )} s 1 - n [1 - Pr(] | > x )]
k=l Xk “ k=l Xk “
n | l
s 2 Pr( > x ).
k=l X1 n
now in a position to prove the following theorem.

Theorem 4.1.
E: {xk} _b_ 3 we 51 independent randogm variables with _a_

common distribution .1 and satisfying (4.10). Suppose that

E X = o and let Bn .22 a sequence g£_positive real numbers

 

such that Bglsn converges in_distribution £9 the unit normal

law. Then

Pr(ISnI>ann)
nPr([XT>ann)

 

+ 1 as n + w,

.93, equivalently,

Pr(ISn|>ann)

Pr( max >x B )
lSkSnlxkl n n

 

+ 1 as n + w,

for any seguence {xn} .2; real numbers such that

xn = 0(nT), r > 0 being arbitrary.

Remark 4.1. If p > 2, the variance of X is finite and we may take

84

En = Cuflh. If p = 2, variance of X may be infinite. In this case
it is known (see for example [11, 130-32, 173]) that there exists a

sequence of constants C , C d 9, such that, as n ~ a,
n n

n Pr(IXI > Cn) m o,

and

n c'2 f x2dF(x) ~ a,
n
|x|<C
n
and we may choose
(4.15) B: = n f x2dF(x) = n U(Cn)’
le<cn
where U(x) = I l yzdF(y) is a function of slow variation [9, 303].
y 5x

Proof of Theorem 4.1.

 

Take 9 > o and denote by E1 and F1 the events

n
(I 2 X,I S e x B ) and (IX,I > (l+€)x B ) reSpectively, i = l,2,...,n.
j¥i,j=1 J n n i n n

Then, we have (cf. (1.4))
(4.16) Pr(ISnI > ann) 2 n Pr(Fi)[Pr(Ei) - n Pr(Fi)].

Now, the Xi's belong to the domain of attraction of the normal
law so that Pr(Ei) w l as n m a. Thus, given 6 > o with l - 26 > 0

we can choose N so large that for all n 2 N

l 1

Pr(Ei) > 1 - 6.
Also,

S >
n Pr(Fi) n Pr(IXI ann)

85

(4.17) s n Pr(IXI > Bn) ~ 0

as n m a (see [11, 128]). Thus, we can choose N2 so large that

n Pr(Fi) < 5 for all n 2 N and hence, for n 2 max(N1,N2), we have

2
from (4.16),

Pr(ISnI > ann) 2 n(l - 26) Pr(IXI > ann)°

It follows that

 

Pf(|S£|>anfi)
(4.18) lim ‘ ' 2 1.
3:3“ nPr(TX]>ann)

For 1 S k S n and n = 1,2,3,..., define random variables

if I | s an
635:“ x" y

if {xkl > yan.

o -1 -1 2
where y is chosen such that x y m o and x y m a as n m a.
n n n n n

n
Write Snn = kflxkn. Then, proceeding as in (1.7) we have

(4.19) Pr(ISnI > ann)
S n Pr(IXI > (l—e)ann)
2 2
+ n [Pr(IXI > yan)] + Pr(ISnnI > eann).

We Shall deal separately with each of the three terms on the right
hand side of the inequality (4.19).
From (4.10), we have

nPr(|XI>(1-e)ann)
nPr(IX|>ann)

L((l-e)ann)
L(ann)

 

= (1_€)-p

 

(4.20) mafia-p- as “P“-

86

Also, for sufficiently large n,

 

 

.2...(|x...n.nnz -(___““<Bn>) (3)0. [M43212
nPr([x[>ann) ‘ Bp yz L(Bn)L(ann)
n n
nL(B ) x p41
n . n
S C( Bp ) (7) ’
n yn

using Lemma 1.1, where C is a positive constant and n > o is

arbitrarily small. Using (4.17) and the fact that x;1y: + w, it

 

follows that
2 2
(4.21) n [Pr(IXI>yan)]
nPr(lX[>ann)

 

+oasn+°°.

Finally, it remains to consider the term Pr(ISnn' > e x Bn)°
n

We note firstly that since E X B 0,

IE xlnl

If x dPr(X
IXISyan

IA

x)l

IA

If x dPr(X
|X|>yan

x)l

IA

-f;anx dPr(IXI > x)

m
fyanPr(IXli> x)dx + yan Pr(IXI > yan)

(4.22)

o<(ynnn)1’°L(yan)>.

using well known prOperties of slowly varying functions [9, 273].
Case 1 - D "2- Let Xn = 0(nT), T > 0. From Markov's

inequality, we have

it"! I II! 1"- II' I ll '1' I II III I '- Iii. | i '14! I ll]- I.I.Il. Illil|

87

4

- -4 4
> S .
Pr(ISnnI e ann) c xn Bn F(Snn)

= c xn 4B n4[n B(x n) + n(n-1)E(X nl)B(x n)3
+ 6(6-1){B(xm)2}2 +

+ n(n-1)(n-2)[E(X1n)}23(xln)2

+ n<n-1><n-2)<n-3)[E<x1n)}“J.

Then, using (4.11) and (4.22), we obtain
>.'
Pr(ISnnl e ann)
nPr(TXI>ann)

L< a n+> ' L(B > [L( B >12
5 C [K: T) 1(3)::n B n)+ 1 2(“ 2 n 7 L(B:)L?ann)

xn BU
(nL(Bn> [M(yn3n>12
B2 L(Bn)L(ann)

 

 

 

,nL(Bn )“2, L(yn B n) 2 M(yn B n)
x2 2 K Bz *7 K_ L(Bn ) n7 ' L(xn B :)

xnn Bn

+——_

 

4
+ 1 nL(Bn))3 [L(yan)] ]

2 .
n

x y: ( B: [L(Bn)]3L(ann)

From (4.17) and Lemma 1.1, it therefore follows that

Pr(IS |>e x B )
nn nn
nPr(]XT>ann)

 

(4.23)

Case (ii). p > 2. Let xn = 0(nT), T > o and take A > 0,
an even integer such that A > p + (2T)-1(p - 2). We have, using Markov's

inequality,

88

rr(|snn| > e ann)

s c x'AB”A (s )A
n n nn
. _ -A -A A A-l
(4,24) — c xn Bn {n B(x1n) + n(n-1)E(X1n)E(X1n) +...}.

Let A = i1 + i2+...+ im be a partition of A into positive integers.

A bound for the corresponding term in the preceding expansion is given by

-A -A 11 i
c xn Bn nmIE(X1n) |...IE(X1n) m|.

Since there are only a finite number of terms in the expansion on the
right hand side of (4.24) it will be sufficient to show that

1 1
-A -A m 1 .m
an Bn n IE(X1n) I...IE(X1n) |

.— —Oo

 

R = -
n nPr(|X| > ann)

as n ~ w for every partition (i1,...,im) of A.
Let u = number of. i. = 1, v = number of ij = p, w = number of

*
ij > p and w = 2 1,. Then (4.15) together with estimates obtained
i.>p
J

in Lemma 4.2 yields
‘ _yA-p *___
RSCnm1(-B) (y8)u+w+PwDuPA.
11 Km nn

[L(ymfim)J“+"[M(yan)Jv
L(x B )
n n

 

Writing Bn = C\/; we see that the exponent of n on the right hand
side of the preceding inequality is

*
u+w +p-wp-up-A

m - l + 2

89
which is
S(l-%)(u+v+w-l),
using the fact that
*
A22m+w +wp-u-2v-2w.

Since p > 2, it follows that (4.23) holds whenever u + v + w 2 1.

If, on the other hand, u + v + w = 0 then all the ij's satisfy
2 S ij < p. The worst case is when all the ij's equal 2 and then
we have
5-1 A/2
2 [M6 5y )1
R s c n “

 

“ (ann)“"p ' L(C ffixn)
np/2-1 [M(c ffiynnuz

x A—p . L(C fix)
n n

 

= C

 

1 [me fay >1“2
n
= C ~ 0 as n a m,

“(A-6544541 me fix“)

 

Since
(A-p)T-Ǥ-+1>o.

This completes the proof of (4.13) in both the cases and the result

(4.14) follows from Lemma 4.3.

Section 4.3. Results on Iterated LogarithmTFy e Behaviour.

Let Xi, i = 1,2,3,... be independent and identically distributed
n
random variables with law iXX) and write 8n = 2 Xi, n 2 1. We
i=1
suppose that Pr(X > o) > o, Pr(X‘< o) > o and that E X = o if

‘..fi 11""

1" .1"! ‘II. 1" ‘ II! All ..Illcl'. ' l‘ .I‘ 1" All II III» is... I I.

90
EIXI < m. If there exists a sequence of positive constants {cn: n 2 1}
such that

_1 _ -
Pr(lim sup on ISnI - 1) - l,

n-wo

we shall say that iterated logarithm type behaviour is possible for

the type X. It is well known that if E X2 =c12 < m then iterated

 

logarithm behaviour occurs for cn =<572n log log n while Strassen
[28] has given a (rather inexplicit) construction due to Freedman to

Show that iterated logarithm type behaviour may occur in a variety of
other cases.
The following result which relatesto this problem has very

recently been obtained by Heyde. Only the statement is given.

Theoreg_4.2.

type X, then-ip'belongs.p_ the domain p§_partia1 attraction

 

.Qi the normal distribution.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11].

[12]

[13]

[14]

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