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LIBRARY
MECMQOII State
University

 

 

 

This is to certify that the
dissertation entitled
SERIES REPRESENTATION FOR PROCESSES WITH
INFINITE ENERGY AND THEIR PREDICTION

presented by

Arnavaz P. Taraporevala

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Statistics

 

l
tic/imam 0L2 gar

Major professor

 

Date August 9, 1988

 

0

MS U is an Affirmative Action/Equal Opportunity Institution 042771

SERIES REPRESENTATION FOR
PROCESSES WITH INFINITE
ENERGY AND THEIR PREDICTION

by

Arnavaz P. Taraporevala

A DISSERTATION

Submitted to
Michi an State University
in partial fu lllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Statistics and Probability
1988

ABSTRACT

SERIES REPRESENTATION FOR PROCESSES WITH
INFINITE ENERGY AND THEIR PREDICTION
By

Arnavaz P. Taraporevala

The purpose of this work is to present series representations for stochastic

processes {X n e 1} whose second moments need not exist. In Chapter I,

n’
we obtain such a representation for SOS processes in terms of c—invariant
exchangeable random variables. For series in c—invariant exchangeable random
variables we associate a dispersion distance and study a prediction problem for
them in terms of minimizing this distance. In case of series in i.i.d. random
variables in the domain of attraction of a stable law our results give those of
Cline and Brockwell. In Chapter II we see that the predictors obtained in
Chapter I are metric projections. In Chapters III and IV we give
nonanticipative series representations in terms of orthogonal random variables.
This problem can be looked at as an orthogonal Wold decomposition in certain
Banach Spaces. The definition of orthogonality is based on the concept of a
semi—inner product introduced by Lumer. Under certain geometric conditions
the uniqueness of the semi-inner product is proved. If the Banach Space is

Lp, p > 1, our results give the recent work of Cambanis, Hardin and Weron

who use James orthogonality.

To Mamma.

iii

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation to Professor Mandrekar for
his guidance and encouragement and to Professors R.V. Erickson, J. Gardiner
and C. Weil for reading this thesis and for their helpful comments. I would
like to thank Cathy Sparks for her superb typing.

I am grateful to the Department of Statistics and Probability, Michigan
State University and the Office of Naval Research for their financial support.

Finally I would like to thank my family for their continued support.

iv

TABLE OF CONTENTS

Chapter
0 Introduction

I Series Representation of Stable Processes;
Dispersion Distance and Prediction

II Metric Projections
III The Left Wold Decomposition
IV The Right Wold Decomposition

References

Page

34
42
61
66

CHAPTER 0
INTRODUCTION

Let {Xn,n E I} be a second order process with EXn = 0 which is

purely nondeterministic. Then Xn has a moving average representation

(D
Xn =k=§m an,k (k. Here the {k's are orthogonal and < Xn’ 5k > = 0,

k > n. In case {X n E l} is a purely non—deterministic Gaussian process

11’
{ék’k e l} are i.i.d. Gaussian random variables. For non-second order
processes with E|Xn|p < oo (1<p<2) similar orthogonal representations are
studied in [3], [5] and [19]. However even in the symmetric a stable (808)
case no distributional prOperties of {éi} are Obtained. In the non—second
order case, it was shown in [18] that there are severe limitations in the use of
the spectral theory. However the use of the time domain techniques for
processes represented in terms of series of i.i.d. random variables lead to
interesting results ([8]). In this approach a suitable dispersion is defined on a
sequence Space through identification Of coefficients of i.i.d. random variables in
the series expansion.

In Chapter I we first consider the relationship to the series
representation of a SOS process in terms of exchangeable random variables.
This allows us to define a dispersion distance on the corresponding sequence
Space which is shown to be a subspace of (a. We then introduce an
apprOpriate dispersion distance On the space of series in exchangeable random

variables. In case the representing random variables are i.i.d. stable we

get that the dispersion distance is the usual to distance and if the

exchangeable random variables are i.i.d. (not necessarily stable), then the

distance is weaker than that given in [8]. Using this distance we study for

ARMA processes with exchangeable input analogues of the prediction results of
Cline and Brockwell ([8]). The technique used in the first problem is an

adaptation of that due to Dacunha—Castelle and Schreiber ([10]) which also
suggests an apprOpriate diSpersion distance. As a by product we also obtain
sufficient conditions for the as. unconditional convergence for the series in
exchangeable random variables. As a consequence we get some results due to
Cline ([7]). In some cases the minimum (p—distance predictor obtained
coincides with the metric projection. This will be seen in Chapter 11.

Let {Xn,n E I} be a SOS sequence (1 < a < 2) and
Mn = §{Xk,k g n}, where closure is taken with respect to the LD norm
(1 < p < a). Cambanis, Hardin and Weron ([5]) have defined concepts of
right and left innovations and orthogonal Wold decomposition using James
orthogonality. Left innovations (which always exist) and orthogonal Wold
decompositions are in terms of {(11, n e I} where {n = Xn - Pn_1Xn and
Pn-l is the metric projection on Mn—l' Right innovations and Wold
decomposition exist in terms of {(n, n E I} if and only if
E[Xn|Xj, j 5 n — l] e Mn—l and in this case (n = Xn - E[Xn|Xj,jgn—1].
However, James orthogonality is not enough when we consider general Banach
spaces. In Chapter III we define the concept of orthogonality for a Banach
Space .3 using the semi—inner product introduced by Lumer ([17]). If
.2” = Lp, p > 1, then Lumer's construction of the semi—inner product ([25]) is
the same as that considered by Cambanis and Miller ([4]). The Lumer
semi—inner product enables us to extend the definitions of right and left
projections as defined by Cambanis and Miamee ([3]) for a general Banach

Space. It is seen in [3] that if {Xn,n E l} is a SOS sequence such that
E[Xn|Xj,j 5 n—l] E Mn—l’ then E[Xn|Xj,j 5 n—l] is the right projection of

XonM

n n—l' In Chapter III we see that Lumer orthogonality implies James

orthogonality. If the Banach space is Lp, p > 1, then Lumer orthogonality
coincides with James orthogonaltiy ([25]). Let x = {xn, n E l} g .3
Mn(x)= i5 {xm, m g 11}, P11 denote the metric projection on Mn and rn
denote the right projection on Mn’ 11 E I. In Chapter III we see that left
innovations always exist if .3 is reflexive, rotund and has a rotund dual.
Further, left innovations and Wold decompositions are in terms of {{n,n e l}

where {n = x — P In Chapter IV we prove that if .3 is reflexive,

n n—lxn'
then the right Wold decomposition and innovations exist if and only if

rn__l(xn) exists for each n. In this case the decomposition is in terms of

{(11, n E l} where Cu = xn — rn_l(xn).

CHAPTER I
SERIES REPRESENTATION OF STABLE PROCESSES;
DISPERSION DISTANCE AND PREDICTION

For a purely non—deterministic Gaussian process {X n e I} we can

11’
choose i.i.d. random variables (fin, n E I} such that {£11, n E 1} forms a
symmetric basis ([16]). In this chapter we first consider the structure of a
symmetric stable process {Xn,n E l} of index a (in short SOS) for which
M 0(Xioo) = s_p'a{Xn,n E I} has a symmetric basis. Here ’0’ denotes the
closure with respect to the norm || [I a defined by (1.1). This motivates
us to study as. convergent series in terms of exchangeable random variables.
We define a suitable dispersion distance on this space and consider the
prediction problem with respect to this dispersion. This extends the work of
Cline and Brockwell ([8]).

For a $018 random variable X with characteristic function

EeitX = exp(—7|t|a), 7 > 0, define

on ”Mia: 71/“ it lsasz

([23]). Then for any 1 g p < a, X 6 LD and

(1.2) llxlla = c(p,a) lenp

where [[XI lp denotes the LD norm of X and c(p,a) is a constant

which depends on p and a ([4]). Hence all Lp norms are equivalent.

Note that I] [la gives rise to a metric and if O > 1, then H [la is a

norm ([23]).
We now start with some basic definitions.

Definition 1.3. A basis {xn} of a Banach space is called an

unconditional basis if every convergent series of the form 2 anxn converges
n

unconditionally. A basis {xn} of a Banach space is said to be a symmetric
basis if it is equivalent to the basis {x 1r( n)}’ for any permutation r of the
integers.

Note that every symmetric basis is an unconditional basis.

Definition 1.4. Random variables {5i, 1 5 i S n} are said to be
exchangeable if their joint distribution function is invariant under permutations
of {1,...,n}. A sequence {(11, n E N} of random variables is said to be an
exchangeable sequence if every finite subset is exchangeable. A sequence
{(11, n E N} of random variables is said to be c—invariant if for every n 6 IN
and n—tuple (k1,...,kn) e In consisting of distinct elements the 211

n—dimensional random vectors (ck {k """k {k ), 6k = t l, have the same
1 1 n n j

probability law.

Let {X n e I} be a SOS sequence (O > 1). By the Kolmogorov

n,
consistency theorem (Theorem 36.1 [1]) we may assume that {Xn’ n e l} is
a sequence on (R1, will”). Since (Rl, 3R1» is a standard Borel Space
and a has no atoms, (“waltz“) is Borel isomorphic to ([0,1], 3[0,1]),/\)

where A denotes Lebesgue measure on ([0,1], 30,1]) ([21] p. 116). Hence

we may assume without loss of generality that {Xn,n e l} is a SOS

sequence on ([0,1], .fl[0,l])). Define
Mp(X=n) = 513'" {ku s n}

Mp(X:-co) = 2 Mp(X:n)
Mp(x:..) = sip {g Mp(X:n)}

where .1) denotes closure with respect to the Lp—norm || [I p if
1 S p < O, ‘0’ denotes closure with respect to the norm || [I a defined by
(1.1) and Sp denotes linear span.

Let {en, 11 E N} be a symmetric basis for M a(X:oo). Following the
proof of Dacunha—Castelle and Schreiber ([10]) we will get for 1 5 p < O a
sequence of c—invariant exchangeable random variables {§n,n e II} in

Lp(O,.9,'P) such that 2 cnen converges in Lp if and only if 2 cnén
n n

converges in Lp(fl,.9,'P). Since (en, 11 6 II} is a symmetric basis for

M a(X:ao), we get by (1.2) that {e n E N} is a Symmetric basis for

n,
Mp(X:oo). Propositions 22.2 and 21.4 [24] imply that {en, n e N} is a

bounded basis for Mp(X:oo), 1 S p < O. Let
(1.5) 0<k Sinf||e|| <sup||e|| 5K <ao.
p nEN n nEN n p p

p"

Since (1.5) is valid for any 1 5 p < O, {en, n e I} is uniformly integrable.

Let 15 p < q < O. Then 7 = q/p >1. Further
K32 Menu?l =1 lenlq = I(len|p)7-

Hence sup E(|en|p)7 _<_ K: < 00. Therefore {[enlp, n E IN} is uniformly
n

integrable.

Let ”ck ,mfik ( or pk ”wk ) denote the probability law of
1 n 1 n

(ekl,...,ekn) for any n E IN and k1,...,kn 6 ll distinct. Let Sn denote the

group of permutations of {1,...,n}. If or 6 Sn let

a(el,...,en) = (0(e1),...,a(en)) = (e0(1),...,ea(n)). Let I‘n denote the group of

multiplication by ({1,...,6n), ‘j = i1, that is ,7 6 PD,

7(e1,...,en) = (7(e1),...,7(en)) with 7(ej) = iej. Define
”n = —l—n 2 2 ”07(8 e )
n!2 a 7 1"”’ n '

For m 5 11 let p: m be the marginal of a: on the first In coordinates.
By (1.5), {#31, n 2 1} is tight and hence ([1], p. 331) has a weakly
convergent subsequence {143 (1)1, nk(1) 2 1} converging weakly to L1.

k ,
Using (1.5) again we see that “11(1),? nk(1) 2 2} has a weakly convergent
subsequence {flzk(2)’2, nk(2) _>_ 2} converging weakly to I12. Continuing in
this manner we get for each me", a probability measure I‘m such that
{”nk(m),m’ nk(m) 2 m} converges weakly to ”m where
{nk(m), nk(m) 2 m} is a subsequence of {nk(m-l), nk(m—l) 2 m} and

{nk(0), nk(0) 2 1} = {1,2,....}. Since ”rim is the marginal of ”rim-PP ”m

defines a sequence {{k, k e N} of c—invariant exchangeable random variables
(by the Kolmogorov consistency theorem) on some probability Space (9,3,?)

Further, by (1.5), {6k} C Lp(fl,.9;P). We will now show that 2 cnen
n

converges in Lp if and only if E cnfin converges in Lp(fl,..9,'P), for any
n

sequence {cn} Of real numbers. But

p _ I I p "
Elk? cIcgkI " £mlclxl cmxml d"m(xl""’xm)
(1.6)

_ - P 3
— 11m £m [c1x1+...+cmxm| dflnk(m),m(x1""’xm)’

k-i oo

Since

P P
|cl cklek1+...+cmckmekm| 5 A(p,m) 'cj‘k.ek.l

ism JJ

sA(p.m)(§ur> chlp); IekI".
15m 15m 1

where A(p,m) is a constant depending only on m and p, and as
{len|p,n e N} is uniformly integrable, it follows that the family
{|cleklekl+...+cmckmekmlp: kl#---#km,(k1,...,km) e um, cj=tl, Igjgm} is

uniformly integrable. Fix m e N. For n 2 m let

1
T = E
n,m n m
m! ( m)2

 

lclrklek1 lp.
lSklf- ° - #kmSn

c ==tl
R.
J

+...+c c e
m km km

The sequence {TIl m,n 2m} is uniformly integrable. We prove this as
follows. Suppose not. Then there exists an 17 > 0 such that for each

6 > 0 there exists a set D and an n > m such that A0(D) < n and

{) Tn,md20 > c. The definition of Tn,m gives

Ice 9 +...+c c e IpdA > c
I!) lklkl mkmkm 0

which contradicts the uniform integrability of [c c e +...+c c e ID.
1 k1 kl m km kIn

The functions Tn m converge in the 0(L1,L°°) tOpOlogy along

2

{n (m),n (m) 2 m} and consequently T dA converge.
k k 0

nk(m),m

(1.7) J (Tnk(m),m-l)d).0

_ P 3 P
._ £111 [C1X1+...+meml dflnk(m),m(xl,...,xm)-’EIkgmckékI .

Suppose X ckek converges in Lp. We now Show that E ckgk converges in
k k

Lp(fl,..9;P). Since {(11, n E IN} is a sequence of c—invariant random
variables, the random variables cnén constitute martingale differences (Remark

2.2.2 [10]) such that E] 2 ck§k|p is an increasing function of II. By the
kSn

martingale mnvergence theorem, 2 cnén converges in Lp(fl,.9;P) if and only
11

if lim El 2 c 5 II) < 00. Since {e , n E N} is a symmetric basis for LP
11 kSn k k n

' P
and 1213 cnen converges In L , we see that the set {1213 cn6n91r(n)}’ where r

runs over all permutations of integers and 5n are scalars such that

|6n| S 1, is bounded ([16] p.53). Hence there exists a constant K such that

10

(1.8) Elcc e +...+c e e [p <KE|c1el+.. .m+c e In|p
1k1k1 mkIn km

for every choice of k1#°°-#km with l 5 kj 5 n for j = l,...,m.

Consequently

8
(1.9) [m |c1xl+...+cmxm[p dun,m(xl,...,xm)
R

<KE|c1e1+...mm+ce|p.

Using (1.7) and (1.9) we see that 2 C11 6n converges in Lp(fl,.9;P).
11
Conversely suppose that E Cnén converges in Lp(fl,3;P). For each m e IN
n
there exist random variables ek ,...,ek such that

m
P D
Elclekl+...+emekm| 5 2E|cl§1+...+ cméml .

Assume otherwise. Then

1

 

I} E|cl e +.. .+c e [p
m!(nm)_nk1mrnk
l<k1:f~ 9! km

P
> 2E|c1§1+...+cm£m|

which contradicts the fact that

11

EIB C-é-Ip =
15111 J]
. 1 P
11;? —(—Tm! ”k m 2m 2 Elcleklek1+...+cmckmekm[
( m ) 15k1#---#km$nk(m)
c =i1
1‘1

. 1 P
= 11m 2 Elce +...+c e | .
k»... m! “klml 15k I...“ Sn (m) 1 k1 m km
( m ) l m k
Since 2 Cnén converges in Lp(fl,.9;P), A = sup E| E ckéklp < 00.
n m kgm
Therefore E|c e +...+ c e | p 5 2A. Using this and Theorem 22.1 [24] we
1 k1 m km
P
see that Ilkg‘lmckekl 'p S 2AK1 < 00 for every m. Therefore i3 ckek

converges in Lp. By Proposition 2.3.8 [10] 2 C151. converges in Lp if and
It

only if i? Elckgklp < 00. But
EElckéklp = {(3 Icklp Elfiklp = E|§1|p E Icklp.

Hence 12‘ ckek converges in Lp if and only if 2 ckEk converges in

Lp(fl,..9;P) if and only if g = {Ck} E [p' Therefore M a(X:oo) is isomorphic
to [p (15p<O) which can be continuously imbedded in (0. Using this

imbedding we can define dispersion(Y1,Y2) = 2 [clan—cg)” where

_ (k) . _
Yk — 2 CH en E Ma(X.oo), k — 1,2.

12

W. Since {Xn’ n e l}, is a SOS sequence there exist
. . O
functions {fn, n E I} In L [0,1] such that

1E (21% AX) ”gun"
—og epr . = .
j_-_-1 ka j=1 11‘] a
([15]). Further if {Z(s): s 6 [0,1]} is an independent increment SOS process

with Eeitz(s) = exp(-e|t|"), then the process {Yn, n e I} defined by
1
YD = [0 fn(s)dZ(s)

is stochastically equivalent to {Xn, n e 1}. Hence we may consider
M a(X:eo) C L“. But every symmetric basic sequence for a subspace of Lp,
1 5 p < 2, is equivalent to a unit vector sequence in some Orlicz sequence
Space ([16] p. 149). This therefore gives a geometric condition on M a(X:ee)
in order that it has a symmetric basis.

We now introduce some definitions and results on Orlicz Spaces which
will be used throughout the thesis. This material is taken from [14] and [27].

For further information the reader is referred to these books.

Definition 1.11. An Orlicz function (p is a continuous, even,
nonnegative function, nondecreasing for positive x such that tp(x) = 0 if

and only if x = 0.

W. A measure It on a measurable Space (0,5) is

separable if there exists a finite or enumerable subcollection .96 of measurable

sets of finite u—measure having the prOperty that if E is an arbitrary set of

13

finite p—measure, there exists to each 6 > 0 a set F E 5;) such that
p(EAF) < 6.

Note that if p = counting measure on (N, power set of ll), then u
is separable.

Let p be a o—finite separable measure on (0,9). We say that two
measurable functions f and g are equivalent if f = g a.e. [p]. Let
.1 = 49,331) be the Space of equivalence classes of measurable functions
determined by this equivalence relation. For an Orlicz function cp and

f E .1! define

(1-13) pap“) = l WW!

and .2; = J¢(fl,3,'p) = {f 6 J: p ‘p(f) < co}. ‘th is not a linear Space in

general.

Example 1.14. Let gp(x) = elxl - |x| — 1. Then p is a convex

Orlicz function such that J") is not a linear space ([14]).

Mfinitiee 1.15. An Orlicz function p is said to satisfy the

A2—condition if there exists h > 0 such that
(1.15.1) ¢(2x) S hgp(x) for x 2 0.
Beka 1.15. If «p satisfies the A2—condition then 'th is a linear

space ([27], p. 81). The Orlicz function in Example 1.14 does not satisfy the
A2—condition ([14]).

14

Exmplfi 1.17. 1) Let (p(x) = |x|p. Then (p is an Orlicz function
satisfying the A2—condition. Further .2"p is the classical Lp space.
2) <p(x) = (1+|x|) ln(1+|x|) - |x| is a convex Orlicz function which
satisfies the Az-condition ([14]). Further 17’ is different from any Lp
space ([14]).

Definitien 1.18. Two Orlicz functions (p and w are said to be

complementary if

(1.18.1) xy S 900:) + ¢(y) for all x,y 2 0.

Let (p be a convex Orlicz function. Then ([14]) (p can be
IXI
represented in the form (p(x) = [ p(t)dt where p(t), the right derivative
0
of cp, is a non-decreasing, right continuous, nonnegative function defined for

t 2 0. Then the function q(s) = Em t is a non—decreasing, right continuous
p t 58

function defined on the nonnegative reals for which q(t) 2 0, t 2 0. If

IXI

I/2(x) = [ q(s)ds, then It is a convex Orlicz function and 1]) is the
0

complementary function of (,0.

Samuel-.12- 1) Let to(x)= IxIp/p. For we

p(t) = <p'(t) = tp_1. Therefore q(s) = sq-l, where % + i = 1. Hence

IS) = Ileq(s)ds = 1%13
0

(2) If p(x) = 6le -|x| - 1, then the corresponding complementary function

18

l5

1/J(J<)= (1 + IXI)ln(1 + IXI) - IXI
([14]). This is an example where (,0 does not satisfy the Az—condition but

its complementary function it satisfies the Az—condition.

Beka 1.29. In many cases it is impossible to find an explicit formula

2
for the complementary function e.g. <p(x) = ex - l ([14]).
For f E 3140,3371) define

(1.21) ||f||¢ = inf {A > 0: [ tp(f/A)dp S l}

and

(1.22) 1‘90 = L‘p(n,.9,'p) = {f 6 Ja- [|f| l‘p < 00}.

In particular we define

(1.23) 1‘|p = L (p(ll, power set of N, counting measure).

Then L (p is a linear space, called an Orlicz Space, and [[0]] (p defines a
semi-norm on L (p' If (p satisfies the A2-condition, then L (p: 'th' If, in
addition, to is convex, then I] | | (p iS a norm (called the Luxemburg norm)
and (L‘p, [l-ll‘p) is a Banach Space.

We now assume that the Orlicz function g) is convex and satisfies the
Az-condition. Let 7) denote the complementary function of «p. If f e L ,

90
then ”mnw < e, where

16

Illflllg, = sup {II fsdu|= gel“), 1),)(3): 1}

(1.24)

= 311D {I Ifsldw gel»). p¢(g) S 1}
and
(L25) Ilfllvs |||f||l¢52 Ilfllw

”lo I | | (p is also a norm on L ‘p and is called the Orlicz norm.

x 1 1.2. Let (p(x) = |x|P/p,1< p < to. Then L‘p is the

classical Lp Space with the usual Lp—tOpOlogy. If f 6 LP, then
1111111,, q Ilfllp and Hill, Ilfllp where p q .

Prepeeitien 1.27. Let (p be a convex Orlicz function satisfying the
A2—condition with complementary function (1;. Then, for any f e L (0’
g E Lw‘)

(1.27.1) l fg d7: 5 lllfl l I,p p¢(s)

Earlier in the chapter we saw that if {X n E l} is a SOS sequence

n,
(O > 1) and M 0(Xfiao) has a symmetric basis, then M 0(Xioo) is isomorphic
to any (p where l 5 p < O. Since [p C (a, 1 S p < O, we can define a

dispersion distance on M a(X:oo) to be the (a distance.

17

Motivated by this let us consider the space

(1.28) Q = {{cn}: E cnfin converges as.)

where Km 11 6 II} is a sequence of c—invariant exchangeable random
variables. Let F denote the distribution function of {1. Assume that F
is not concentrated at 0. It will be seen later that this is not a stringent

condition. Define
°° 2 2

(1.29) (p(x) = I (x u Al)dF(u), x E R,
o

where xAy = min(x,y) for any x,y E II. Then cp is an Orlicz function

satisfying the A2—condition. Let g = {cn} E [W Then

(1.30) n; Pllcn§n|>ll + n31 E cfitfillcnénl s 1]

= 023 E(c2£2A1) = 2p (9) < co.
n=1 1‘ n 9”
Let Yn = cnfin [|cn§n| 5 1]. Since [190(9) < co,
00 W
(1.30)' n21 P(cn§n at Yn) = nil P(|cn§n| > 1) < 00.

Hence, by the Borel-Cantelli lemma, P(cn§n a! Yn i.o.) = 0. Therefore,

18

co
2 c “5 converges a. s. if and only if 2 YD converges a. S.. We shall
n=1 n=1

now prove that 2 YD converges a...s Let 5;] = a{§k, k 5 11}. Since
n=1

3'
E|Yn| 5 1, E "-1 Yn exists and by the c—invariance property of
{£11. n e N}. A Y]l dP = A (—Yn) dP for all A e 3&4. Therefore,

‘9n-1 2
E Y =0 foreach nell. Inviewofthefactthat I3 EYn <00

n n=1

(by (1.30)) and by PrOposition IV.6.1 [20], 2 YIl converges a.s.. Therefore,
n:
00
2 c Mg converges a. 3.. Thus we get the following result.
n=1

I '_'-I I I. a ( ), ( ) ( ),
J C

Remegk 1.32. If in addition the random variables {(11, n e N} are
independent, it is known [‘p [C (see, for example, [2]).
Suppose that {{n, n E II} are c—invariant, exchangeable random

variables . Suppose {cn} is a sequence of real members such that 2 cnén
n

converges a.s.. Let xn = engn, x“) = xn [|xn|< 1],x1(12)=xn - xfll).
Then {X£1), X1512), m, n e N} is a sequence of c—invariant random
variables. Further, as {511, n E l} is assumed to be an c—invariant

exchangeable sequence

19

In

2 X

(1°33) Xxn’ Xn + Xn+1""’k=0 n+k)

= axgl’mfiz)XX“’+X(2)+X(11+Xfiir-nkr§0(X(it+xn+b>

_ “(1),,(2) x11)x(2),x(1)-x(2),, ”20041-22”

foreachfixed m20, n21. Let

S = 2 x 3(1):: 2 x19), 3(2): X(2).
n k—l k’ k

Thenforany m5n and (>0,

II M):

k

(1)_ (1) .
le’é‘iién'sk Sm '> I

= P[ max |2(S(l)—S(1))+(S(2)—S(2))-(S(2) —s(2))) > 26]
m5k5n

< P[ max [(319) —S(1))—(S(2) -S(2))|> >c]

m5k5n

+ P[ maxn |(s(1) -S(1))+(S(2) -s(2))| > 6].

Using (1.33) we get for any n 2 m

1.34 P max S(l)-S(1) >c 52P max S —S >6.
( ) [m5k5nlk m I 1 [m5 Snlk m| ]

But [ max |s(1)—s(1)| > r] T [sup|s(1) —s(1)| > c] and
m5k5n

[ max [Sk—Sm I > c] I [supISk-Sm | > c]. Letting n -1 oo in (1.34) we get,

m5k5n

20

1.35 Psu S(l)—S(1) > c 52P su S—S > c.
( ) [k2rIf1I k m l l [k2rIfII k ml ]

Suppose {81(3)} diverges with positive probability. Then there exists 6 > O
and 6 > 0 such that for every m fixed

65P su S(l)—S(l) >c 52Psu S—S >6
[1121ng n m I ] [“211]?n ml 1

so that {Sn} diverges with positive probability. This contradicts the fact

(1) - °° (1)
that {Sn} converges a.s.. Therefore {Sn } converges a.s. 1.e. n21 X n

converges a.s.. Further sup|X£l)| 5 1 so that supIXIgl)| 6 L2. Let
n n

.2 - a{§°k<n} and O -{2E%‘1(x(1))2< } B Pro Sition
n’ n' - o- n n °°' y po

IV.6.2 [20], 2 X511) converges a.s. implies P(flo) = 1. Therefore

co on
D; E cit: llcnénl s 1) :31 1104,”)2

co .7 co .2
= )3 E E m40(0))? = E[ 2 E n“1(x(1))2]19 .
n=1 n n=1 n 0

If, in addition, {(11, n E N} is a sequence of i.i.d. random
. 2 2 . °° .
variables, 2 E c 5 A1 < oo 1.e. 2 (p(c ) < oo. If further {6 , n E IN} IS
11 n n n=1 n n

(D
a sequence i.i.d. SOS random variables with O > 1, then 2 (p(cn) < oo
n=1

. . a . . _ _
if and only 1f :3 [cu] < 00. Therefore In this case If — (1p — (a.

21

Q
Lemme 1.3§. Let c = {c } E l . Then 2 c E converges a.s.
— n (p n=1 n n

no
unconditionally. Further nEI c ”6 21121 c1005 1(a) as. for every

rearrangement {x(n)} of {n}.

(I)

Pr_0(_)£. LOIS YD = c1161] “Cngnl S 1]. Let n21 CT(H)€T(H) be a.

rearrangement of the series 2 c ”5 But
n=1

M8

,2 -P[Yr(n) * cuisine] = Pllcnn)‘r(n)' > 1]

n=1

:3 P[|cn£nl >11< .. (by (130)).

Hence 112151105110 ) converges as. if and only if 11le 1'( ) converges a.s..
yn—l
Let ‘21.: a{£l’(k), k S 11}. Since ElYfln)l S l, E Yfin
all A e 5:1'1,
551:1 2 2
E Y -0.By(1.30)21EY =n21nEY <00. Hence,by
1r(n ) n_ 1r(111)

) exists and for

Since [ Yr(n) dP = I (-Yr(n)) dP, we get

(D

Proposition IV.6.1 [20], n2 lle) converges a.s.. Therefore c11(n)£7r(n)
n=1

converges a.s.. Hence 2 c “g converges a.S. unconditionally. Let 5' denote
n=1
the tail a-algebra of {(D, n E N}. Let {7r(n)} be a rearrangement of {n}.
m

m
L = , '= o = ... 00° °
et Sm jg] Y1, sm 1.211%). Qm {1(1), ,r(m)}A{1, ,m} By

Corollary 5 in §7.3 [6]

22

2 2
ms -s') = E Ev.+E EY.Y
m m J'EQ J iZREQ 1“

wk
= E Ev? + 2 E EJYXk
160 J i.k€Q J
2 #k at a 2
= E EY. + E E(E .)(E )= 2 BY.
160 J LREQ J k jeq J
#k
. 2 w 2
If {7r(1),...,7r(m)} D {l,...,j}, then E(SIn — 8&1) S E E Y. -+ 0 as

k=j+1 J

j-+oo. Hence S -S' 40 in probability. But Sm 4 EYn a..s and
m m n-l

(D I
-+ 2 de ) a...s Hence 2 Yn = 2 Y “0 ) a.s.. By (1.30) we get

n=1 n=1 n=1
00

E, C an6 ‘a-(nmn)

n1n=

m
Notation 1.37. For a = {an} E (C and X =n21 anén, define

pspoo =21 «:(an).

The minimization problem considered will be with respect to this
translation invariant distance.

Let {{n,n E I} be a sequence of c—invariant exchangeable random
variables. A Special type of process which iS a moving average in c—invariant
exchangeable random variables is the ExARMA process given by the stochastic

difference equation

(1.38) X - 01X

11 a Xn—p = {n + 016n—1+"'+0q€n—q

n—1_. . ._ p

with transfer function 9(z)/A(z), where

23

(1.33.1) em = 1 + 012 +...+ 0qzq

(1.38.2) A(z) = 1 — alz —...- apzp

for z E Ii = {z E C: |z| 51} are such that
(1.38.3) 9(z)A(z) # 0 for all z E I).

If

00
k
(1.33.4) fig} :20 «k2 , z e If (note x0 = 1),

no
then Xn =k20 ark {n—k’ the convergence being a.s.. Note that {Xn}

defined this way is a stationary sequence. Conversely suppose that "0 = 1

m
and {In} 6 [80' Then Xn = 1‘20 ”kgn-k’ convergence being 3.3., 18 a

stationary solution of (1.38) with transfer function 9(z)/A(z) satisfying

(1.38.4). Note that in view of Remark 1.32 in the special case when (i are
Q

i.i.d., E [jg—j converges 3.3. if and only if {wk} 6 l‘p. AS observed earlier
i=0

one does not necessarily get this condition for non-i.i.d. random variables. We

now prove extension of results obtained by Cline and Brockwell ([8]).

Remark 1.39. 1) Suppose {uj} is a sequence such that

00 In

00 m
mil p(jgl Vj’m—j) < 00. Then by Lema 1.36 “1251.31 Vj’rm—j)£n+l—m

24

co

1 2 2 . = 2 .X ..
converges a.s. unconditionaly and m=l (j:luj1rm_1) (n+1—m j=1 VJ n+1—j

2) Suppose 2 (p(uj) < co, 2 (0(uj) < 00 for some n e I. Then
j=n+1j=-oo

2 VH6 and 2 ujjfi converges a.s. unconditionally (Lemma 1.36).
j=n+l J J wj=—ooJ

Therefore Y =.E VjEj converges a.s.. For Y of this form define
J=oo

10¢“) =j§_m ‘P(Vj-)

Thggrem 1.49. Let {Xn’ n E I} be the process satisfying (1.38) —
(1.38.3). Let S... be the class of random variables of the form

1.40.1 Y = 023 6 E X -
( ) j=—n+1jéj + j--- —1 ”j n+1-J

where 2 (p(dj)<oo, 2 (p(jBV xm_ )<oo. Foreach Y€S*,define
j=n+l m=11uj j

=j{21aX

j n+1

2 1(p(21 Vjfl'm_ j) < co and

(1.40.2)

p (Y - 23 an is minimum}.

)
(,0 j—l n+1—j

Then P ”Y consists exactly of one element; namely,

*

*
Y = )3 VJXn +1—j‘ Furthermore the mapping Y -+ Y is linear on 8*.
j=l

25

00
Jim. Let Y E S... be of the form Y =j=§+15j£j +j§1ijn+1-j'
Then, by assumptions, Lamma 1.36, and Remark 1.39

v— zlajxn+1_j= 2 5535 +2(u. —a.)X

j= i=1 i=1 “1‘1
co
= jEI 6j€j +m§1(j§11’(Vj—aj)xm—j)§n+1—m'
Therefore
pr-j21a= 23 6j + 2 21 ij1rm—a)_)

(1.40.3)

2 2 90(6)

j=n+1

If V]. = aj for each j, then

1),,(Y-j3a )= 2 so(5j-)
la_an+1j j=n+1

Conversely suppose there is an equality in (1.40.3). Since «,0 is an Orlicz

function,

2(1A-aj)m_j=0(mEN).
j=l

If m = 1, then 0 = (VI-a1)ar0 = Vl—al and hence Vl -

— a1. Suppose
Vm = am for m=1,...,k. Then

0 " (”1‘3‘1) 1rk+1—1+°"+("k+1“"k+1)"(1<+1)-(k+1) = Vk+1‘ak+1

so that Vk+1 = ak +1. Therefore, by induction on m, Vm = a for all m.

26

Hence p “(Y -2a laj Xn+1 -j) is minimum if and only if um = am for each m
* *
1.e. PY- {211/an+1 -j}' If Y =j§l ijn+l—j’ then the map Y-oY

is linear on 8* (by our assumptions and Lemma 1.36).

r 11 1.41. Let {£11, 11 E I} be i.i.d. random variables satisfying
*
(1.38) — (1.38.3). The the map Y -» Y is continuous at 0 with respect to

convergence in probability in both spaces.

P rmf Suppose Y(k)= Eblk)§j + 2 u(k)Xn +1 65* for each
j= =n+1J j= —1 J je

k are such that Y(k) -9 0 in probability. Then

Y(k)= 2 6905. + 2: (.2 VCR):

j=n+lJ J m=1 j=11m_j)€n+l-m.

By Corollary 2.35 [2], 2 rp(6(k)) + 2 p(j 2 l1/(1‘)1rm__.) -» 0 as
j= —n+l m=1 m]

k 4 co. In particular 2 gp( 2 u(k)1rm_.) -+ 0 as k —. oo.
m=l j=1J 1

Using Corollary 2.3.5 [2] again we get

Y(k)* =2 u(k)x .=°§ (2 14“):

40
J1J n+1—J m=1j= 1J

xm—j)€n+l—m
*
in probability. Therefore the map Y a Y is continuous at 0.

figmgk 1.42. Suppose {(D, n E l} are i.i.d. random variables which

00
are not necessarily symmetric. Let <p1(x) = I |ux|AldF(u) and
-00

27

<p2x( )= I m11112x2A1dF( ). Then (p1 and (p2 are Orlicz functions satisfying

co
the A2—condition. Let b = {bn} be such that 2 w1(bn) < on. Then as
n=-oo

(D
902(bn) 5 cp1(bn) for all 11, 112-00 «p2(bn) < co. Further

2 Elbn {n |[|bn En |_>_ 1] < E cpl(bn ) < 00. Hence, by the Kolmogorov
n:

--00 n=-oo

three series theorem 2 |bn 5n | converges a.s.. Instead of S,., in Theorem
n-«»

1.19 consider S* which is the set of random variables of the form

(I)
Y=2 (SHE-+2 V-X

1:.“ n ,-_ 1 firm-J

where 2 cpl(6j) < co, 2 1<p1(. 2 uxm__ ) < 00. Then, as in Theorem 1.40,
j=n+l m=1 j=1 J "j

={021u X
1:1

J n+1—j} and the map Y -1 2 VX . is linear on S*.

j-l J n+1—J
The proof of the following theorem is similar to Theorem 2.2 in Cline

and Brockwell ([8]) and hence is omitted.

Thmrem 1.43. For the ExARMA (p,q) process {Xn, n E l}
satisfying (1.38)—(1.38.3) there exists a unique minimum gp—dispersion linear

prediction X1: +k for X k 2 1, based on the infinite past Xn, Xn—l’

n+k’
This predictor satisfies the recursive relationship
* 10-]. (I)

where l - wlz — ¢222- = A(z)/8(z), z 6 ll.

28

kmark 1.44. Suppose one has only the data Xn,...,X1. For
Y 6 8*, of the form (1.40.1), define the truncated predictor by

Y (n) =j§1VJXn+1—j' By the equality in (1.40.3) we have
on
p WY-Y (11)) - p,p(Y-Y ) = p WY-Y (11)) - 3 945,-)
j: —ln+

* 00
But Y—Y (n) = 2 6.5. + E ( 8 V. _.) 5 _ .
j=n+l J J m=n+l j=n+1 J arm J n+1 m

Therefore
Ill *
p¢(Y-Y (11)) - p,p(Y-Y )
co m
= 2 50(6j)+ 2 <p( 2 V.1rm_.)— 2 <p(6j)
j=n+1 m=n+l j=n+lJ J j=n+1
co m
= 2 «x 2 m_,-)

m=n+1 j=n+1 VJ

*
In part1cular, 1f Y = Xn+1,then ”(p(xn+1—Xn+l) = p(l) and

*
Xn+l ‘ Xn+1(n) = 6n+1 +j_ 1213+ 'ijnH—i

= 6n+1 +m_§+l{j_1§l+lijm—an-l-l—m'

Therefore

(xn+1—x;,l(n)) = sp(1)+ °2° «x ’3 wj«m_j)

p
“’ m=n+1 j=n+1

so for large n the truncation is nearly Optimal.

29

Let Xn satisfy

(1.44.1) X - 01X

11 0X =5Il

n_1 -. . .- p n_p

andlet n>p. Then ¢j=0 if j2p+l sothat
*
and hence the truncated predictor is optimal. Assumption (1.38.3) reduces to
A(z) at 0 V z 6 ll.
We state the following lemma and corollary whose proof is similar to

Theorem 1.40 and Theorem 1.43 respectively.

Lemma 1.45. Let Xn = (Xn’Xn—l""’xl)' Let S,..(n) be the class of

random variables of the form Y = Z + 1_/' En for some K E R“ and

(D (D
Z = 2 6.5. such that 2 <p(6.) < 00. Then, for each Y e S*(n), the
j=n+1 J J j=n+l 1

set an = {£1}an p (p(Y - a’Xn) is minimum} consists exactly of one

variable. For Y = Z + g'Xn, this unique variable is Y = V’Xn.

Furthermore, the mapping Y 4 Y is linear on 8*(n).

Corollary 1.46. For the process (1.44.1), provided n > p, there exists

a unique minimum predictor X n +k for Xn +1‘(k 2 l) in terms of X1,...,Xn.

This predictor satisfies the recursive relationship

A A

Xn-l-k =‘aan+k_1+...+ apxn+k—p
with the initial condition Xj = Xj, 1 5 j 5 n .

Let {5D, 11 E N} be i.i.d. random variables with the prOperty that

there exists 0 < a < 2 such that

30

, P(|£1l>tx) _a
(1.47) 11m W "'—" X for each X > O.
1

t-+ao
We now prove a result of Cline ([7]). In order to do so we define regularly

varying functions and state a theorem from [12] (p. 275—281).

Definition 1.4g. A positive function defined on (0,ao) varies slowly at

infinity if for each x > 0,

th

(1.48.1) lim t

t-loo

=1.

A positive function U defined on (0,00) varies regularly with exponent p if
and only if

(1.48.2) U(x) = xpL(x)

where -00 < p < co and L is slowly varying.

Thmrem 1.42. a) If U varies regularly with exponent 7, then

p+1
(1.49.1) Wep+7+h p+7+120
P

where

t
(1.49.2) U (t) = J pr(x)dx.
P o

31

b) If L varies slowly at infinity, then
1“ < L(t) < t‘
for any fixed 6 > 0 and all t sufficiently large.
Suppose {5n,n e N} is a sequence of c—invariant exchangeable
random variables satisfying (1.47) for some 0 < a < 2. Let (p be defined
by (1.29) and let

(1.50) L(t) = taP(|5l| >1), U(t) = t_aL(t).
By (1.47), L is slowly varying at infinity and U is regularly varying index
—a. By Theorem 1.49a)

2
W42—a sothat
1x

(1.51) U1(x) - x2(2—a)‘1 U(x) = x2(2—a)P(|51| > x).

Since 2(p(a) = Ea2521A1
= P(|a£1|>l) + E3262llla€1|slh
(p(a) 2 P(|a51| > 1) = U(T%l-) for each a.

Q
But 2(p(a) = Eazgfm = J P(a25%Al>t)dt
o

1
= J P(a25%A1>t)dt
o

l
gJHJfixmt

o
1llal
=2], |a|23P(|51| >s)ds

32

and hence (0(a) 5 2|a|2 Ul(-[%r) so by (1.51) there exists a constant K
such that
1
10(3) S KWW) = KP(|3§1| > 1)

for |a| sufficiently close to 0. Therefore
(1.52) P(|a51|>1) < (p(a) < KP(|a51|>1)
for |a| sufficiently close to 0. Theorem 1.4%) gives us

(1.53) egg)“ < Ial‘“ P(la€1|>1) < (1217“

for c > 0 fixed and all |a| sufficiently small. (1.52) along with (1.53)
imply there exists constants K1, K2 > 0 such that
Kllala'i" < (p(a) < K2|a|a_‘

for c > 0 fixed and la] sufficiently small. Thus if _a = {an} E (p for

00
some p < a, then 2 an5n converges unconditionally and
n=0
co co
n20 anén =m20adm)5 1r(m) for any rearrangement {7r(m)} of {n} (by

Lemma 1.36).

Let {5!}, n E I} be a sequence of i.i.d. random variables, which are
not necessarily symmetric, satisfying (1.47). Let L and U be as defined in
(1.50). Let (p1(a) = E|a5|Al. Then

33

(01(a) = E|a51|A1 = ([QP(|a5|Al>t)dt
l
= (I) P(|a5|A1>t)dt

1
S (I) P(|a5|>t)dt

= 11 ( t r“ L(t/|a|)dt
0 ET '
Let |a| be sufficiently close to zero. Hence for any 6 > 0

_a(

Tfi)‘ dt.

1
= |a|G-€ I tC-‘(I d8
0

1 t
W1“) 5 (I) (131')

Let (>0 besuchthat c+l—a>0. Then

aO—C aG—f
W1<al5jai$=i¢i$°

Since cp2(a) = Ea25fiAls 1pl(a), we have

(1.54) 902(3) 3 ma) 5 {$5.

Let 0<6<1Aa and a={an}e(6. Let 5:0—6. Then c>0

00
and c+1—a = a—6+1—a = l-a>0. By (1.54), E (p2(an) < co and

n=-oo

m 00
2 (pl(an) < 00. Hence by the Kolmogorov three series theorem 2 lan5n|
nz—m n=-oo

converges. This result is due to Cline ([7]). Prediction problem for a e [6 ,
6 < Mo was considered by Cline and Brockwell ([8]).Observe that the

dispersion distance used by Cline and Brockwell ([8]) is an appropriate distance.

CHAPTER II
METRIC PROJECTIONS

In Chapter I we considered a minimization problem in Orlicz sequence
spaces. In a certain class of Orlicz spaces the minimum cp-dispersion linear
predictor is the metric projection considered by Cambanis, Hardin and Weron
[5]. In order to define metric projection we need some concepts from the

geometry of Banach spaces ([13], p. 342]).

Definition 2.1. A function (p. R -+ R is said to be strictly convex if

‘PUX‘i'U-MY) < MOO!) + (l-AMY)
for all A 6 (0,1), x,y E R.

Definition 2.2. A Banach space (xl | - | |) is said to be rotund if it
satisfies the following prOperty: if x,y e .5 are such that x 9% y and

l
llxll = llyll =1, then Ham)” <1.

Definition 2.3. Let (in | ~ I I) be a normed linear space, N g z
x E .1; Define 9N(x) by

50(11): {1' E N: ”H II = inf llx-y||}-
N 0 0 yEN
If 9N(x) consists of exactly one element, denoted by PNx, then PNx is

called the metric projection of x on N.

Remark 2.4. x e N if and only if PNx = x.

34

35

Example 2.5. Suppose that {511, n E II} is an symmetric basis for a
subspace .3 of Lp, p > 1. Let N C .2” be a closed subspace and suppose

N = 5 {5nk, k e S} where S g l is a finite or a countable subset of N.
Let Y E .2: Then there exists a sequence of scalars {an} such that
on
Y = 2 an5n, the convergence being with respect to the LD norm. Let
n=1

*
Y = Ban 511 EN. Then
kES k k

*
(2.5.1) ||Y—Y HP 2 inf llv—znp.
ZEN

11
Let Z=Eb5 EN. Let Y(n)=2 5,Z(n)= 2b5,
1168 “k “k 11:1ak 1‘ nkSn “k “k

Y*(n) =nk§nankgnk, Y(O) = 2(0) = v*(0) = 0.

Then Y(n) - Y*(n) = Y(n) — Z(n) if n < n1.
(2.5.2) i.e. ||Y(n) — Y*(n)| (p = ||Y(n) — Z(n)] (p 11 n < n1.

Since {5D, 11 E N} is a basis ([24] p. 58), there exists a constant
K 2 1 such that

(2-5-3) K ”Y(nl) " Z(n1)l Ip 2 ”Y(nl-l) ‘ Z(fl-IN IP
= ||Y(n1—l) — Y (n1—1)||p (by (2.52)).
*
= ||Y(n1)" Y (I11)||p
Suppose 111 < n < n2. Then as {5D, 11 E I} is a symmetric basis,

36

n

(2.5.4) Kl lY(n) - man I, 2 ||Y(n1-1)-Z(n1-1) + k=§l+1ak€k|lp
= ||Y(n) - Y*(n)l I,
(2.5.5) K ||Y(112)-Z(ng)llp 2 II W23 (a,—b,) c,I I,

l
naén2

= Hung—1) - Y*(n2—1)I I, (by (2.24))
= ||Y(n2) - Y*(n2)l I,

Continuing in this manner we obtain

(2.5.6) K ”Y(n)—zen I, 2 MW!) — Y*(n)l I,.
Letting n -+ 00 we get

(2.5.7) KIIY—ZII,2 IIY—Y*II,.

This is true for any Z 6 N. Thus
* *
llY-Y II S inf KIIY-le S KllY-Y II-
p ZeN * p
In particular if K = 1, then ||Y—Y [I = inf ||Y—Z|| . Therefore the
P ZEN P

*
metric projection of Y on N exists and is equal to Y .

The following prOposition gives us conditions when PNx exists.

Proposition 2.9. Let .5 be a Banach Space
a) Then .5 is reflexive if and only if for every x E .5 and for every closed

subspace N of .5 9N(x) 1: (6.

37

b) If .5 is reflexive and rotund, PNx exists for each x e .5
For proof we refer the reader to Corollary 2.4 [25] and to §2.6.2 (3) [13].

Bemegk 2.7. PN is continuous, bounded and idempotent but not
necessarily linear. For example let .5 = Lp[0,1], 1<p<2, M = {afz a 6 IR}

Where f = 1(o,1/2)’ f1 = 1(0,2/3)+ 1(0,1/4)’ f2 = 1(0.1/3) " 1(0,1/4)’
3 = fl + f2 = 1(012/3) + 1(011/3)' Let x<p> = (sgn x) |x|p for any

x E R, PM fk = akfk, k=l,2, and PM g = af. By Theorem 1.11 [25],

1 1
J f(g—af)<p—1>d/\ = o = J f(fk—akf)<p_1>d,\, k=1,2.
o o

:3, 1(0 1/2)“(0 2/3) + 1(101/3) " a1(0 1/2)) D‘”

so that a = 1+2

:1].

1+2

a
II

1(fl—alf)<P‘1>d1

H

1(01/2)(1(02/3) + 1(01/4) ‘ a111,(01,2)> b“

l/4(2— —a )<P‘1>d1 + 1/2 (1—al)<P‘1>dA

N H O“ CH“ CE
H

=+[(2-31)<p_1> (1—31)<p_1>]

38

so that a1 = 3/2.

—:<‘H>( + ,r) + (2—a2)

so that a2 = —'1_ .Therefore

,9.

1+2?-

+f2) 1+2
=1(0 1/2)

3 2
i 2 1(0.1/2) + ‘1‘— 1(0.1/2)
571”“
f

However the following properties are true ([5]).

Promsition 2.3. Let .5 be a reflexive and rotund Banach space and
N a closed subspace of .5
3.) Then PN(ax) = aPNx for all scalars a and for all x e .5
PN(x+n) = PNx + PNn for all x E .5 and all n e N.
b) If further N has codimension one in .5 then PN: .5 -» N is a linear

operator.

39

Eromeition 2.2. Let (p and 1,!) be complementary convex Orlicz
functions satisfying the A2—condition. Assume further that tp is strictly
convex. Let N be a closed subspace of L SP and let f e L (pnNc' Then the

metric projection of f on N, namely PNf exists.

Pgoof. In view of Proposition 2.6 it suffices to show that L (p is

reflexive and rotund. Since 1;: and 1]: satisfy the A2—condition, L 90 is
reflexive ([27 ,p. 154]). Further since (p is strictly convex L $0 is rotund
([22])-

The following prOposition shows us that if (p is convex, the minimum
tp—diSpersion linear predictor obtained in Theorem 1.40 is the metric projection

with respect to the distance II. | | (p in [,p.

Promeition 2.10. Let (p be a convex Orlicz function satisfying the

AZ-condition. Let N c L w and 1 e L SpnN“ be such that

9Nf = {g E N: p¢(f—g) is minimum} consists of exactly one element, denoted

~

by f (p. Assume further that

(2.10.1) a f = (of) for any a E R.
‘P ‘P
Then PNf exists and PNf = f 90'
Prmf. For any g E L let

‘P

-l
= : < .
Tg {A>0 p,,(/\ g) - 1}

40

Let f EN and AeT . By (2.9.1)
1 HI

-1 -l
p,,(/\ (HQ) 5 9,,(3 (f-fl)) S 1

so that A e Tf_f¢. Hence Tf—fl g Tf_f¢.Therefore

= . —1 '

[If-f‘pl Jtp 1nf {p¢(,\ (f—f‘p)). A E Tf—ftp}
. -1

S 10f {P900 (‘41)): A E Tf__fl} = llf’fll '56

This is true for any fl 6 N. Hence, as f cp e N, we get

Therefore PNf exists and PNf = f‘p.

Qorollegy 2.11. Suppose that {5n,n E l} is a sequence of

c—invariant, exchangeable random variables. Suppose the distribution function

F of 50 is not concentrated at the origin. We further assume that the
(D

Orlicz function tp defined by (p(x) = J u2x2A1dF(u) is a convex function.
0

Assume the conditions of Theorem 1.40 are satisfied. Define
- _ m . _ .
N — {{bm}m=_mel(p. bk — 0 1f k 2 n+1 and
n+1—k

bk = j§l aj'n+l—j—k otherwise}.
L if 6 01% s d l * ‘2 x
t. Y = . . .X . E t, Y = . ..
e j=n+l 1&1 +j=1 V “H * an e j=lVJ “1'3

Let a = {ak} and 2* = {a;} be defined by

41

 

6k 1: E {n+1,n+2,....}
3k =
n+1—k .
j-El Vj Irn+l_j_k otherwrse
* 0 k E {n+1,n+2,...}
3k =
n+1-k
E Vj’n+l-j—k otherwise.
. i=1

 

:1:
Then PNa exists and PNa = a .

* -
W. Note that a E (II? and a E N. Further by Proposition 1.40
- .1. :I: :1:
9~2={a}and(oe) =03
N

*
for any scalar 01. Hence by Theorem 2.10, PNa exists and PNa = a .

Bemark 2.12. 1) If (p(x) = |x|“, 1 S a < 2, then the dispersion
predictor of Cline and Brockwell ( [8]) is the metric projection in (a.
2) Note that in Example 2.5 we consider {5n,n E R} be a symmetric basis
for a subspace .5 of LD and find conditions for the existence of metric
projection with respect to the LD distance. In Corollary 2.11 we have seen
that for a certain subset S... c ((p, the metric projection exists, the metric here

corresponds to the Luxemburg norm in [W

CHAPTER III
THE LEFT WOLD DECOMPOSITION

Let {X n E I} be a stationary second order process with

n,
E XI] = 0 for all n. Then the moving average part of its Wold
decomposition is constructed as follows. Let Mk = 3‘15 {Xn’ nSk}, and

5k = Xk — PMk Xk’ k E l, where PM denotes the projection on M. Let
—1
M_m= 2 Mn ={0}. Then observe that {5k, k E I, k 5 0} is a basis in M0

and Sn5k = 511+k where SII is the shift Operator on M00 = 5% Mk}

co
given by Snxk = Xk +11. From this we get X0 =k2 ak5_k and
=0
00

Xn = 1‘20 akén—k' We note that [51,: k E I, k _<_ 0} is a symmetric basis

for M0 and in case {Xn: n E l} is a Gaussian process {5kz k E l, k g 0}
are i.i.d. random variables.

If {Xn} is a symmetric a stable (SaS) process with a > 1,
Cambanis, Hardin and Weron ([5]) have used the concept of James
orthogonality to define left and right Wold decomposition and innovations. In
this chapter we extend these concepts to general Banach spaces using a
semi—inner product introduced by Lumer ([17]). This gives rise to a definition
of orthogonality. We will see later (Proposition 3.14) that Lumer orthogonality
implies James orthogonality. In case the Banach Space considered is Lp (p>1)
the two definitions of orthogonality coincide.

We now introduce the concept of a semi—inner product following Lumer
([17]) to extend the definitions of right and left projections defined in [5]. The
Definition 3.1 and Proposition 3.2 are taken from [17]. Here F denotes the

field of real or complex numbers.

42

43

kainition 3.1. Let .5 be a vector space over F. A semi—inner
product is said to be defined on .5 if for any x, y E .5 there corresponds

aelement [x,y] in F with the following properties:

(i) [x+y,z] = [x,z] +[y,z] for all x,y,z E .5
[Ax,y] = A[x,y] for all x, y E .5 A E F.

(ii) [x,x] > 0 for x at 0

(iii) llx.y]|2 s [x,x] Iy,yI for an x, y e x

Proooeition 3.2. Let .5 be a normed linear Space over F and let
5“ denote its dual. For each x E .5 there exists Wx E 5 such that
Wx(x) = (x,Wx) = ”XII? and ”wa = [|x|]. For x, y e .21 define
[x,y] = (x,Wy). Then [~,] defines a semi-inner product.

kmgk 3.3. Suppose 5", the dual of .5 is rotund. Let x E .5
By the Hahn—Banach theorem there exists Wx E .5“ such that
2 * * . *
(x,Wx) = [|x|*| and ||Wx|| = ||1:||. Suppose x1 1% x2 are 1n :5
such that ”x,” = [|x|] and (x,xk) = ||x||2, k = 1,2. Since .2; is

l * *
rotund ||§(x1 + x2)” < ||x||. But
I * :1: 1 :1: *
(x. 20‘1“») = §(xl+x2)(x)
_l * 1 * _1 2 2 _ 2
- 5x10.) + 2X26) — glllXII + ”X” l — ”KH-
31' II!
This contradicts the fact that ||%(x1+x2)|| < ||x| |. Hence for any x E .5
there exists a unique element Wx E .5 such that wa” = ||x|| and
(x,Wx) = ”XI |2. Therefore the semi-inner product is uniquely defined in this

case. Note that in a Hilbert space the semi—inner product is the inner product.

44

W- Let .5 be a reflexive, rotund Banach space such that
its dual .2? is rotund. Let x e .5 and M be a closed subspace of .2:
Then PMx, the metric projection of x on M, is uniquely determined by
[y, x—PMx] = 0 for all y E M.
We find the form of Wx for Orlicz function Spaces. For this we

need extension of [14] (p. 73 and p.88).

W- Let tp be a convex Orlicz function, let p be its
right derivative and 11) be the complementary function tp. Suppose (p and
to satisfy the A2-condition. If f E L (,0 then p(|f|) E L ¢ and if

lllfl I I, 31,
(3.5.1) p¢(p(lf|)) s I I lfl I I,

met Let 5i={EE5;p(E)<co}. ForanyEEyl,
p(|f|)lE E L . Further

IllflEIIIS, = sup {1] lfgldw 36%,. 1),/Is) S 1}

s sup {llfgldw 1565,. 1),,(3) S 1}

That is,
(3.5.2) IllflElll s IIIfIII,

If p(|f|) = 0, then (3.5.1) is trivially true. Now assume that
E0 = {x: p(|f|)(x) J! 0} 9% 4). Let |||f| | '90 S 1. We now Show that for
each E E .9i, p¢(p(|f|))lE) S I||f| | |(p. Suppose this is not true,

45

i.e.,suppose there exists F E 51 such that p¢(p(|f|)lF) > |||f||| 90' Then
FnEosttfi. For xEFnEO,
¢(DIf|(X)) = “p(lflbt») < ‘P(f(x)) + Wp|f|(X))= |f(X)|p(lf|(X))-
Therefore
P¢(P(lf|)1F) < l lflpflfl) lep
s I I lflpl I I,o,(p(IfI)1F) (Proposition 1.27)
s |||f|||¢0¢(p(lfl)lp) (by (35-2))
which contradicts the fact that |||f| | | (p 5 1. Therefore for each
E e a, p,/,(p(lfl)1F)<|||f|||,,- Define V(E) = woman.) for each
E E .9.’ Then V is a a—finite measure. Further V(E) S |||f| | | ‘p for each

E E .7 . Therefore sup V(E) = a 5 |||f| || . Hence there exists a sequence
1 E63 ‘9

{E} in .9',E T suchthat limV(En)=a. Let B=UEn. Then
11 1 n D400 11

a = V(B). If E0 E .5, we have

I) (p(|f|)) = p (plfll ).<. |||f||| '
to e E, to
so that (3.5.1) holds in this case. Now suppose that E0 9! .9i. Let

E, = E0 n BC. Let F e .71, F c El. Suppose V(F) > 0. Then

a = V(B) < V(B) + V(F) = V(BUF)= lim V(En UF) < Eszup'yV(E)=

11400

which is a contradiction. Hence V(F) = 0. Since V is a—finite this implies
that V(El) = 0. Thus as BC = E u (BC n E3), we have

..(BC): V(EI) + u(BC n EC: o—.
Thus P¢(p(|f|)) = P¢(P(|f|)13) + p¢(P(lf|)ch)

= V(B) + ”(30) = as |||f|||,
which proves (3.5.1).

46

W. The following proposition gives another formula for
evaluating | | | . | | | cp‘

Emeition 3.7. Let (p and 16 satisfy the conditions of the previous
* *
theorem. Let f E L ‘p and suppose kf = k is a positive number such that

 

(3.7.1) o,(p(k*IfI)) =
Then
(3.7.2) IIpr(k*IfI)on = |||f|||,-

Egoof. Let 1* be defined as above. Then

*
l|f|1>(k lflldu S 811 llfgldfl = |||f|||,-
P¢(g 51

0n the other hand, as 1p and 11) are complementary functions,

|||f| I I, = E: :an IIfIn(k*IgI)on
IIMR fldfl + I 10(3)an
5 El? klfl:(k: f) + 1]

= i” Ip,(k*r) + p,(n(k*IfI)>1

= i” Ik*IfIn(k*IfI)on

*
= l|f|p(k |f|)d/t s |||f|||,

which proves the result.

47

W. Let ((3,111 be as in Pr0position 3.7. Let [-,-](p be a
semi—inner product defined by the norm |||~ | | | 10' Let f, g E L (p be such

that k; exists. Then

IIIgIII, = Ilglp(k;|g|)du = Ig(sgng)p(k;|gl)du.

Hence

[is], = Ifl l Isl l l,(sgng)p(kglgl)du

W This example shows us that the inner product defined by

Cambanis and Miller ([4]) is a particular case of the semi—inner product defined

here. Let (0(z)= J—J— (l<a<2). For x > 0,

tp'(x) = p(x) = xahl. Let )6 > 0 be such that 1

a+%=1. Let 10 be

6
the complementary function of 1;). Then (p(y) = lfi-L (Example 1.19.1) so

forany k>0

o,(nIkIgI)) = }, IInIkIngdA = 71,1 (k|g|)fl("_l)dk

a
= %, ((k[g|)°’d,\ = %— Hg”:-

Hence p¢(p(k|g|)) = 1 implies

_ka a. _ l/a . *_ la—l/a
-B—llgllar 80 that k—ngg—n'z, 1.e. kg— 8 a

 

Hence ntk,IgI)= (1;th) ”sh, 1=|—|§||—7Ig gl

 

48

By Example 1.26. lllgl I I, = WIIsII, and hence

* __ 1/B 2-a a—l
Illslll, p(kglgl) - 6 Hall, Isl .
Thus

1st = WI IgI If“ I ,,<oe1>,,,

where g<a'_1> = sgn(g)|g|a_l. Therefore

 

EAL? = lfg<a_l>dp.

a
lllglll, Ilgl on

We now assume that (.5 | | - | |) is a Banach space over F with
:1:
rotund dual space .5. Let [~,] be the semi-inner product defined by the
norm ||- | |. The following definition extends the concepts of right and left

projections as defined by Cambanis and Miamee ([3]).

Definition 3.13. Let (.5 | | - | I) be a Banach space over F and let
[-,-] be a semi—inner product defined by the norm ||- | |. Let M be a
closed subspace of .5 and let x E .5 11 MC. The right (resp. left) projection
of x on N is defined as an element r(x|M)(resp.l(x|M)), of M satisfying

(3.10.1) [x,y] = [r(x|M),y] for all y E M
(resp.

(3.10.2) [y,x] = [y,!(le)] for all y E M).

Here [-,-] is the Lumer semi—inner product.

49

W. Let M be a linear subspace of a Banach space .5
and let x 5 3n NC. If r(x|M) exists, it is unique.

m. Suppose 71, 72 E M are such that
[71,y] = [x,y] = [72,y] for every y E M. By definition 3.1(i) we get
[71-72,y] = 0 for every y E M. In particular, as 71—72 E M,
o = [71-72, 71-72] = ||71‘72l I2 so that 71 = 7, (Definition 3.1).

Definition 3.12. Let (.5 | | . | I) be a normed linear Space over F
with semi—inner product [-,-]. Let x,y E .5 x is James orthogonal to y,

denoted by ery if
(3-12-1) IIX+Ay|| 2 IIXII
for all A E F. x is said to be orthogonal to y, denoted by x1y, if

(3.12.2) [y,x] = 0

Let 5, .52 be two subspaces of .5 511.1% (resp. 521.15) if x11x2 (reSp.
xlaJx2) for each x1 E 5 and x2 E ..g.

Beka 3.13. 1) By the form of the semi inner product in LP,
1 < p < co, xny if and only if xtJy, ( Theorem 1.11 and Lemma 1.14 [25]).
This is not necessarily true in general. However if my then X1 Jy which

will be seen in Pr0position 3.14.

50

2) If .5 is a Hilbert space then by [25] (p.91), my if and only if x1 Jy if
and only if <x, y> = 0 where <~,-> is the inner product.

3) Note that my (resp. x.1Jx) does not imply that y1x (resp. y1 Jx). For
example let 1 < p < 2, f1 = 21[0’1/4) + 111/411/2) — 3ll3/411l and

f2 = l[0.1]' Then f1, f2 E Lp. Let x<p> = |x|p sgnx.

_ <p—1>
[f21 f1] "' [fl f2

= (1/4) (29‘ + 1 — 31”) > o
_ <p—1>
and [f1,f2] — I f2 f1
= ’ (”Ion/4) + l[1/4.1/2)" “Ia/4,11)
= 1/4 (2+1-3) = 0.
Therefore f21f1 but f1 is not orthogonal to f2. By the previous remark

f21Jf1 but f1 is not James orthogonal to 1'2.

Browsition 3.14. Let .5 be a normed linear space over F and
x,y E .5 If my, then XLJy.

goof. Let A E F. Since my, [y,x] = 0. Hence
1st = 1an + AIynoI = [X+Ay.x}-

So
(3.14.1) [x,x] = |[x+Ay,x]| 5 [x+Ay, x-l-Ay]1/2 [x,x]1/2.
If [x,x] = 0, then x = 0. Hence ||x+Ay|| = [IAyII 2 0 = ”X”; i.e.

xtJy. Let [x,x] > 0. Then from (3.14.1) we get

51

1/2 1/2 ,

[x,x] S [x+Ay, x+Ay] , i.e. ||x|| 5 ||x+Ay| |.

Therefore x1 Jy.

Bromsition 3.15. Let .5 be a normed linear Space. Suppose {xn}
is a sequence in .5 converging x E .5 Let y E .5

a) If y1xIl for each n, then y1x.

*
b) If xn1y for each n, .5 is reflexive and .5 is rotund, then XLy.

PM. a) If y = 0, then y1x. Now assume y#0 so that
My” > 0. Let c > 0. Since xn -+ x, there exists n0 E I such that
n 2 n0 implies ||xn — x|| < JE/I [y] I. But ynxn; so [xn,y] = 0. Hence
[x,y] = [x-xn, y]. Therefore |[x,y] |2 _<_ [x—xn, x—xn][y,y] < c for
n 2 no; i.e. |[x,y]| < c. This is true for any 5 > 0. Hence [x,y] = 0.
b) Let Wxn’wx be elements of .5 corresponding to xn, x E .5 (cf.

Pr0position 3.2 and Remark 3.3). Since xn -+ x, there exists M > 0 such

that
(3.15.1) ||WX H = ||xn|| 5M for each 11.
Il
*
Further wa l] = ||xn|| —. ||x|| = ||Wx||.Since .5 is reflexive, .5
n

is reflexive ([19], p. 135). Hence by (3.15.1) {W x } has a weakly convergent
n

subsequence {Wx }. Without loss of generality assume that Wx
n n
k

Ik *
converges weakly to an element x E .5 . Then

* . . 2
(3-15-2) IX (X)| = 11111 IWx (X)| S 11111 IIWx || IIXII = ”KM
11

Il-ioo 11 than

52

2 .
But ||xn|| = Wxn(xn) = Wxn(xn—x) + Wxn(x).Since
IWxn(xn-X)l S IIWxnll IIXn-Xll S M IIXn-Xll -' 0.

:1:
wx (x) -. x (x) and |[xn||2 .. ||x||2. We get
I1

(3.15.3) [|x| (2 = [0.)

*
(3.15.2) and (3.15.3) together imply that ||x [I = |[x| I. By Remark 3.3,
*
x = Wx. S1nce xn1y, 0 = [y,xn] = Wxn(y). Hence

[y,x] = W (y) = lim W (y) = 0; i.e. my.
X 11400 xn

*
Let .5 be a Banach space over F with rotund dual space .5 and
5, .52," be closed linear subspaces of .5 We now define a concept of an
orthogonal (1) decomposition for general Banach spaces. For certain class of

Banach spaces orthogonal decompositions where considered in ( [5]).

Definition 3.13. The symbol 5+...+51 .51) denotes the

H

A
o
H
has
ll MI:

subspace {x1+...+xn: ij.5J,1$k$n}. 51-55%" .51) denotes the

y—s

A
o
'1
he
ll M8

subspace '33 U ( means

11 J 1
.5: 5+...+51 and

II MI:

.51). .5: 5 3....0 .51‘l(or

..J

i.“
ll MI:

J—I
1.6

B.
V

(3.16.1) 5+...+5( 1 fi+l+m+5l for all 1 5 k < n.

53

Writing .5: “fl 0....6 .2; means that .5: fi+...+c%;l and
a.

s.

(3.16.2) $+...+ J. +...+ for all 1 5 k < n.
n "fiwl “fit "q

(D m m
.3: 2 $ .$ (resp. .3: 2 9 .3) means .3: 2 .5 and (3.16.1) (resp.

(3.16.2)) holds for all n.

ark .17. 1 Note that e = 6 . Also,asnoted in
M ) 51415 «£31

Remark 3.13.1, the statements .2” = .fi 6 .32 and .3 = “fl 0 5 are, in
4 (-
general, distinct.

as
2) Let 5:2 $3. Let0#x.E$. Forany mSn and
jzlql J J

fl1,fl2,...e F we have by the definition and Lemma 3.14 that
’3 E
. . < . . .
l|j=1fljlel - ”1:1 51"."

Hence ([24], p. 54) {xj} forms a basis for its closed linear Span, i.e. each

x 6 a5 {sz j = 1,2,...} has a unique norm convergent expansion

on
x = 2 ijj for some A1, A2,...EF. Note that the same argument cannot be
i=1

00
made when .3: E 0 .3.
j=1 (-

We now consider Wold decomposition under definition of orthogonality. For

this we need the following proposition .

54

*
W. Let .5 be a reflexive Banach space such that .5
is rotund. Suppose there exist subspaces .5; and Ln of .5 such that

(3.18.1) .5: .5I'l 0 Lne....6 Ll for each n 2 1,

c- t- 4-

Then

(3.1s.2) .3: 33 (eLn)c(n.5;1)
n=1 -9 -O n

Prggf. Let .3100 = n .Zl'l, Kn = Ln 0...0 L

n o—e—

By the definition of 6, K 1.5 for each 11. Since .5 C .5, for each n,
.- n n -00 - n

l and Km='s$(;Jl Kn)'

K15 foreach n.Letk€UK.Then3nEIl suchthatkEK.
n -uo n n n

Hence,as Kn1.5 ,[x,k]=0 for any xE.5 . But kEK was
1:0 ~00 n

arbitrary. Therefore U Kn“$—oo' Let k 6 Km. Then there exists a sequence
n

{kn} in 3 K11 such that llkn-k|| -o 0 as n —+ 00. Let x E .510. Since
{kn} Q U Kn, [x,kn] = 0 for each 11. By Proposition 3.15 b), [x,k] = 0.
m

This is true for any x E .5_m and k E Km. Hence Km15_m. Thus in order
to prove (3.18.2) we need to show that .5 = Koo-+500. Let x e .5 Then
x = xn + kn where Xu 6 .55, kn e K n’ Since kn‘an’ knlen (Proposition
3.14). Therefore

Ilknll S llk,1 + xnll
so that ”an = ||x-kn||

IIXI l;
2| |x| |. Hence the sequences {xn} and {kn}

IA

are norm bounded. Since .5 is reflexive, they have simultaneously weakly

55

convergent subsequences {xn } and {kn } with weak limits x and k
j - -'m 00
respectively. Hence x = x_m + km. Since .500 is a closed subspace of .5

it is convex. Hence .500 is a weakly closed subspace of .5 Therefore

x_m e .5_m. Further there exists a subsequence {ym} such that

ynj E co{kn1,...,knj} = convex hull {kn1,...,knj} and ||ynj- km” -) 0. Hence
k e K . Therefore .5 = K 0 .5 .
°° 0° 00 -m

.9

Remark 3.12. From Remark 3.17.2, we recall that any km 6 Km has

00
a unique norm convergent expansion k = 2 kn’ kn 6 Ln for each n.
n=1

Let us observe that ([25], p. 111) if .5 is a reflexive, rotund Banach
Space and M is a closed subspace of .5 then PMx, the projection of x

on M, exists for each x E .5 and satisfies

(3.20) ||x-P X” = inf ||x—y||.
M yEM

a:
We now show that if in addition .5 is rotund, then (x—PMx)1M. Note
It 1k
that by the Hahn—Banach theorem ([25] p. 18) there exists x E .5 such that
ill *
“X I] = ||x—PMx||, x (y) = 0 for every y E M and

* *
x (x—PMx) = ||x-PMxl |2. In view of Remark 3.3, x = w Thus

x—PMx'

PMx is uniquely determined by the equation

(3.21) [y, x—PMx] = 0 for any y E M.

56

W. Notice that x = PMx + (x-PMx) and
(x — PMx)rPMx. We want to show that this is a unique representation.
Suppose there exists x1 6 M and y1 E .5 such that

x = PMx + (I—PM)x = x1 + yl and ylrM.
Since yllM, we have by Proposition 3.14 that ylrJM so that for any
y E M.

llx-yll = ||y1+(x1-y)|| 2 Ilylll = llx-xlll;

i.e. ||x—x1|| = inf ||x—yl |.
yEM
Since PMx is unique, x1 = PMx. Thus x 2' PMx + (I—PM)x is a
unique representation of x as a sum of an element of M and an element of
.5 orthogonal to M. In particular mM if and only if PMx = 0.
2) Let Q: .5 -o M be an Operator (not necessarily linear). Suppose
(I—Q).5J.M. Let x E .5 Then x = Qx + (I—Q)x = PMx + (I—PM)x. But
Qx E M and (I-Q)x.LM. Hence by Remark 3.21 PMx = Qx. But x 6 .5
was arbitrary. Hence PM = Q. Conversely suppose PM = Q. Then by
(3.21), (I—PM)mM. Thus Q = PM if and only if (I-PM) .51M.
3) If the Banach space considered is a Hilbert space H with inner product
<-,->, then my if and only if <x,y> = 0 (x,y E H). In this case PM
is linear for every closed subspace M of H and satisfies (3.21) for any
x E H ([25] p. 57). Then by the above remarks H = M 0 M‘ for any closed
.—

subspace M of H where
M‘L = {y E H: <x,y> = 0 for all x E M}.
Following [5] for x={xn,n e I} g .5 a Banach Space we now give the

left Wold decomposition.

57

Define
(3.23) Mn = M(x:n) = E {xkz k 5 11} (past and present of {xn})

(3.24) M00 = M(x:ao) = s; {UM(x:n)} (time domain of {xn})
n

(3.25) MW = M(x:-oo) = n M(_x:n). (remote past of {xn})
n

x = {xn} is said to have left innovations if for each n there exists a

subspace Nn(x) = Nn so that

(3.26) M(x:n) = M(x:n—l) 9 Nn(x).

Notice that Nn(x) is necessarily one or zero dimensional. x = {xn} is said
to have a left Wold decomposition if there are subspaces N n = Nn(_x),

- on < n < 00, 811011 that
(3 ) M( - ( E N M -00
.27 x:n ... o x 6 x: .
) 0 n—k( )) (_ )

Bromitign 3.28. Let .5 be a reflexive, rotund Banach space with a

*
rotund dual space .5 . Then {xn} has left innovations.

film. For the sake of convenience let Pn denote the metric
projection onto M(x_:n). Since the codimension of M(x:n-1) on .5 is one,

Pn—l: M(x:n) 4 M(x:n—l) is linear. Hence Nn(_x) = (I—Pn_1)M(x:n) is a

58

linear subspace. In view of (3.21), Nn(x_)rM(x:n—l). Thus in order to complete
the proof of this prOposition we need to prove that
M(_x_:n) = M(_x_:n-l) + Nn(x). Let y e M(x:n). Then
Pn_1y E M(x_:n—1) and y — Pn—ly = (I—Pn_l) y E Nn()_r). Further
y = Pn__1y + (y—Pn_ly). Therefore M(x:n) = M(x:n—l) 6 Nn()_().
g...

Ngtation 3.29. For the sake of convenience let Pn denote the metric

projection Operator onto M(x:n).

Thflrgm 3.30. Let .5 be a reflexive, rotund Banach space with
*
rotund dual space .5 . The following are equivalent:

(i) {xn} has a left Wold decomposition.

(ii) Pn: Mac -+ Mn are linear.

(iii) The operators Pn: Mm -. Mn commute.

(iv) If Pn,m denotes the restriction of PD on Mm’ then for
all k 2 1’ Pn,n+1Pn+1,n+2“"Pn+k—l,n+k = Pn,n+k’

Pm. We will show that (iv) _. (ii) 4 (i) -+ (iv) and (ii) H (iii).
(iv) -» (ii) Assume (iv) holds. By Proposition 2.7 b) each

is linear. Hence by (iv) P is linear for each k 2 1 so

Pn+l,n+l+1 n,n+k
that PD is linear on each Mn+k' Since P11 is continuous, Pn: Mao -. Mn
is linear.

(ii) —. (i) Assume each Pn: Mm -) Mn is linear. Define Nn = (I—Pn_1)Mn.
Let 2n 6 Mn. By (3.21) ZnLMn—l and thus znan—l for l 2 1. Then

Pn—l 2n = 0 (by Remark 3.22.1). Since Pn is linear, we have

59

Pn—k(zn + zn—1+"’+zn—k +1) = 0. Hence using Remark 3.22,

k—l

Nn +...+N *Mn—k‘ Therefore Mn = ([20

n—k+1 an—I) f Mn-k' By

Proposition 3.18 we get (i).
(i) -» (iv) Suppose {xn} has a left Wold decomposition. Then for all n

0....0 N

-+ -»

n+1SMn soany yEMn+£

n+tf Nn+l—1

is uniquely expressed as y = 211+! +zn+(_l+...+zn+1 +yn where zj 6 NJ.

and yn 6 Mn' Further P But

P

n,n+l y = y1:“
n,n+1"'Pn+l-l,n+Z(Y)

= Pn,n+l‘"Pn+l—l,n+l(zn+[+"'+zn+l+yn)
P

n,n+1"'Pn+l—2,n+l—l(zn+[—l+"'zn+l+yn)

= Pn,n+1(zn+l+yn)=yn=Pn,n+l(Y)
and this proves (iv).

(ii) -» (iii) Assume (ii) holds. Let x 6 M00 and m 5 n. Then
PmPn(x) = Pm{x—(x—an)}
= me — Pm(x—an) ( as PIn is linear )
= me (Remark 3.22).
But m 5 11 implies MIn g Mn so that me 6 Mn‘ Hence
me = Panx. Therefore Panx = Pman. Thus Pn: Mco -9 Mn
commute.
(iii) -. (ii). Suppose (iii) holds. Since each PR is continuous, it suffices to

show that each P11 is linear on each M In view of Pr0position 2.7 a)

n+k’
we need to show that each P11 is additive on each Mn+k‘ By Pr0position

2.7 a), P Let

is additive on Mn“ Assume PD is additive on M
= (I-P

n n+k—1'

x1, x2 6 Mn+k be arbitrary. Let yj = P

n+k-lxj’ Zj n+k-1)"j’

60

j = 1,2. Note that ZjiMn+k—1° By PrOposition 2.7, Pn+k—l is a linear
Operator on Mn+k° Thus
Pn(xl+x2) = Pn(y1+y2+zl+z2)
= Pn+k—1Pn (3’ 1+y2+z1+22)
= Pn Pn+k_1(y1+y2+zl+zz) (by (iii))
= Pn (y1+y2)
= Pny1 + Pny2 (induction assumption)
= Pn Pn+k—l(x1) + Pn Pn+k—l(x2)
= Pn+k—l anl + Pn+k—1an2 (by (iii))
= Pn x1 + an2
and this proves the result.

Remgk 3.31. The above Wold decomposition was proved in [5] for the

case Lp, p >1.

CHAPTER IV
THE RIGHT WOLD DECOMPOSITION

In this chapter we will discuss an extension of the right Wold
decomposition introduced by Cambanis, Hardin and Weron ([5]) by using the
right projection (c.f. Definition 3.10) and Definition 3.16. Throughout this
section we will assume that .5 is a Banach space over F with rotund dual

*
space .5. For this we need the following proposition.

Broggsition 4.1. Let .5 be a reflexive Banach space over F.
Suppose there exist closed subspaces .5; and Lu of .5 with
.5: .5; 9 Ln ewe L1 for each n 2 1. Then

-) -) -D

Q Q
.5: (.2 9 Ln) 0 (n 51) and each k E 2 0 Ln has a unique norm

1:1 +- e— n j=1 ..
co
convergent expansion k = 531“" [n 6 Ln“
11:291. Let .51” = g .51, KD = Ln 3...: L1 and Ken = sp {ii Kn}.

Then, by a proof similar to the one in Proposition 3.18 (we use Proposition

3.15 a) instead of PrOposition 3.15 b)) .5 = .5_no 0 Km. So in order to
...

complete the proof of the theorem it remains to show that each R 6 Km has
a unique norm convergent expansion k = IE1 In, In 6 Ln. For each n we
can write k = X11 + kn uniquely (follows from the definition of orthogonality)
with xn E .5n and kn E Kn' In turn we may write kn = (1 +...+!n

uniquely with (j E L Define Qn: K0° -) Kn by an = kn. Then

j.
Qn Q! = QnAl . Also by Proposition 3.14.

61

62

llanll = ”1"an + kll
SHR*QN|+|WH=IHJI+HHI
S len + knll + ||k|| (by orthogonality)
s 2 Ilkll-

Therefore, by the uniform boundedness principle, {Qn} is a bounded sequence

([9], p. 98). Let k E U Kn' Then there exists n0 E II such that k E KB
11 0

so that k E Kn for each n 2 no. Hence an = k for each n 2 n0.

Therefore s—lim an = k for any k E U Kn' Let k E Km and (>0.
n-+oo 11

Let sup ||Qn|| S M. Then there exists k E U K such that
n C n n

||k—k£|| < c/(M+1). Further there exists n6 E II such that kc E Kn for

each 11an sothat anc=k foreach nan. But 11ch implies

C
llk-anll = llk-kc-(an-anc)“

Sllk-kell + llinl Ilk-kcll < 6-

n 00
k = s—lim an = s—lim 2 l. = 2 (j.
.___1

n-loo n—m j=l J j
Prgmgitign 4.2. Let .5 be a Banach space and M a closed
subspace of .5 Let lM = {y E .5: [y,x] = 0 for all x E M}. Then 1M
is a closed subspace of .5 Suppose further that for each x E .5 the right

projection Of x on M, r(x|M), exists. Then .5 = M elM.
..p
{1293. Let yl, y2 E lM, al, 02 E F. Then for any x E M,

[alyl + a2y2, x] = al[yl,x] + a2[y2,x] = 0 so that aly1 + a2Y26 J‘M.
Hence J‘M is a subspace of .5 By PrOposition 3.15, lM is closed.

63

Let .5 be a Banach space such that r(le) E M exists for all x E .5
Then by (3.10.1)

[r(XIM), y] = [x,y] for each y E M,

i.e. x—r(x|M), y] = 0 for each y E M.
Therefore x - r(x|M) E lM. Further as x = r(le) + [x - r(x|M)] and
as r(le) e M, 5: Me‘M.

4
For any sequence )_r = {xn} in a Banach space .5 let M(x:n),

M(x_:ao) and M(x:-oo) be as defined in (3.23), (3.24) and (3.25) respectively.

Following [5], we say that {xn} has right innovations if for each 11 there is

a subspace Nn = my such that M(x:n) = M(§:n-l) :0; Nn()_c). Note that

Nn(x) is necessarily one or zero dimensional. x = {xn} is said to have a
right Wold decomposition if there all subspaces N n()_(), -oo < n < 00, such

that

m
M(_x:n) = kEO 0 Nn_k()_r) 0 M(x_:-oo), M(§:n) .l. Nm(x_) for each
= c- 4-

Q
m > n and further each z E E 0 Nn_k()_c) has a unique norm convergent

k=0 +-
on
expansion 2 = RED wn_k, wj E Nj(x).
Thwrem 4.3. Let x = {xn} be a sequence in a reflexive Banach

space. The following are equivalent.
(i) x has right Wold decomposition
(ii) x has right innovations
(iii) rn(y) = r(yan_l) exists for each n and for each

y E U M(x:n).
n

64

m. We will show that (i) .. (ii), (ii) ._. (iii) and (ii),
(iii) -» (i).
(i) -o (ii). This follows from the definition of right Wold decomposition and
right innovations.

(ii) 4 (iii). Suppose _x has right innovations. Let y E UM(x_:n). Then there
11

exists n E I such that y E M(x:n). The definition of a right projection and
PrOposition 3.11 imply that y = r(y|M(x:m)) for all m 2 11. Let n < m.
Then there exists yn E M(xzn) and zj E Nj()_(),

n + 1 _<_ j 5 m, such that y = yn + zn+1 + zn+2 +...+zm. Note that each

zer(x:n), n + 1 5 j g m. Hence, for any 2 E M(ggn),

[y,z] = [yn + zn+1 +...+zm,z] = [yn,z] + .351.” [zj,z] = [yn,z].
Therefore r(y|M(x:n)) exists and is equal to yn.
(iii) -’ (ii). Suppose (iii) holds. Let Ln—l = {rn_l(y): y E Mn}' Let
yl, y2 E Mn’ 01, a2 E F. Then, for any 2 E Mn—l’
[alrn—l(yl) + a2rn_l(y2),z]

= allrn_1(y1). Z] + aglrn_1(y2).z}

= allylazi + agiYgaZ]

= [(alyl + ang): z]

= [tn—1(“1y 1 + “23%)”
Since alrn_l(y1) + azrn_1(y2) E Mn—l’ by the definition of a right
projection and PrOposition 3.11, we have that
alrn_l(yl) + a2rn_2(y2) E Mn-l' Hence Ln—l is a subspace of Mn—l‘
Conversely suppose y E Mn—l' Then y = rn_l(y). Thus Mn—l g Ln—l'
Therefore Mn_1 = {rn_l(y): y E Mn}. Let

65

NH = Nn(x) = {y - rn_l(y): y E Mn}' By Proposition 4.2, Nn is a

subspace of Mn and Mn = M e Nn. Therefore _x has right

n+1 _,
innovations.

(ii), (iii) -» (i). Now assume (ii) and (iii). Then for each n,

Mn. ={lrn _1(y)= y E M n}.

NI, :n{y-r _=1(Y) yeMn} and

MD M 1H O NH == (Mn_26 Nn—l) 0 Nn.

-i -o -+
We now show that
Mn = (Mn—2 f Nn—l) 3 Nn
=Mn-2S(Nn—16Nn)= M--n20Nn
...}

Let y1 E Mn—2’ y2 E Nn-l’ y3 E Nn' Then there exist z1,z2,z3 E Mn such
that y1 = ’n-2(‘n—1(Zl)): y2 = l'n-2(‘in—1(22)) ‘ ’n-1(Z2)’

y3 = Z3 - rn_l(z3). Since Nn—l _C_ Mn—l and Mn _lLNn, Nn MIN

Further Mn— 2*Nn—1 But

Iy2+Y3ayli = [3’233’1] + [Y33Y1]
= [rn_2(rn_l(z2)),y1] _ lrn_1(z)a 3'1] + [z3,y1] "' [rn_1(z3)a 3'1]
= 0

(definition of a right projection and as yl E Mn—2 _C_ Mn—l)‘

Therefore M a N 9 Nn' Hence Mn = M 9 Nn

n_10N .
....

n—2 n—2 —10n
.g

Continuing in this manner we get Mn = Mn—k 3 N 0 .. 3 Nn’ Using

n-k-l-l _’
this and PrOposition 4.1 we see that x = {xn} has a right Wold

decomposition.

Remark 4.4. In view of Theorem 3.30 and 4.3 this extends the work of
[5]-

10.

11.

12.

13.

14.

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