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This is to certify that the

thesis entitled

EQUIVARIANCE IN COMPOUND DECISION PROBLEMS
AND A STABILITY OF SYMMETRIFICATIONS OF
PRODUCT MEASURES

presented by
Jin-Sheng Huang

has been accepted towards fulfillment
of the requirements for

Ph.D. degree inflitifitics & Probability

ji’WiJééL VIA/la 4 L

E Major professor

Date February 24, 1970

0.169

LIBRA '{Y

Michigan State
Universi ry

 

EQUIVARIANCE IN COMPOUND DECISION PROBLEMS
AND A STABILITY OF SYMMETRIFICATIONS OF
PRODUCT MEASURES

BY

Jin-Sheng Huang

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

1970

TO MY PARENTS

ii

ACKNOWLEDGEMENTS

I wish to express my sincere thanks to Professor James
F. Hannan who suggested the area of research and whose partic-
ipation at each stage of the development actually amounts to
co-authorship. I am also grateful to Professor Dennis C.
Gilliland for reading the manuscript and for suggesting improve-
ments. I am indebted to my former colleagues for their en-
couragements during my M.S.U. years. Finally I wish to thank
Mrs. Noralee Barnes for her excellent typing and her cheerful

attitude in preparing the manuscript.

iii-

TABLE OF CONTENTS

Part Page
a EQUIVARIANT PROCEDURES IN THE COMPOUND DECISION
PROBLEM WITH FINITE STATE COMPONENT PROBLEM
Summary .. .............. .......... ............... 1
1. Notations and History ........... ................ 2

2. Equivariant Decision Procedures and
Symmetrifications in 3 Compound Decision

Pr0blem I. ..... O ....... ......OOOOOOOIO ........... 7
3. Representation of Equivariant Risk .............. 10

The Difference Between the Two Envelopes When

0 is Pairwise Non-Orthogonal ......... 13
S. The Difference Between the Two Envelopes When

‘9 may have some Pairwise Orthogonality ......... 16

b A STABILITY OF SYMMETRIFICATIONS OF PRODUCT MEASURES

WITH FEW DISTINCT FACTORS
O. sumary O. ...... O ...... .... OOOOOOOOOOOOOOOOOOOOO O 20
1. Introduction ......OOOOOOOOOOOOOO ...... .... ...... 21
2. PrOperties of Symmetrification and an ia-Norm ... 23
3. Permutations; Contraction Effect of Probability

Factors .....0.........OOIOOOCOOOOOOOOOO. ........ 27

4. Two Distinct Factors with Unit Differences in
Mu1tiplic1ties ......OOOOOCOOOOIOOOOOOO 0000000000 29

5. m1+ 1 Distinct Factors .......... ............. .. 37

PART 3

EQUIVARI'ANT PROCEDURES IN THE (DMPOUND DECISION PROBIEM
WITH FINITE STATE COMPONENT PROBLEM

0. SUMMARY

Let (IMB,P) be a probability measure space for each

pea={F

0,...,Fm},<7 be an action Space and L be a loss function

defined on I X 9 X a such that for each i,
= j‘ v L(x,Fi,a)dFi(x) < no.
-a
In the compound problem, consisting of N components each

with the above structure, we consider procedures equivariant under

the permutation group. With

. _3/2 _ am
pm. BZB‘Fim) - Fjam and up) = 1—1 (5)9(19) ,

we show that the difference between the simple and the equivariant

envelopes is bounded by

(TI) {2K(P) E c?}% N“35 where p = V p,
i 1 1.3

and by

-%

0T2) 2m{2K(p) E c:}% N

. where p = V{pij'pij < 1}.
1

The bound (T1) is infinite unless the F1 are pairwise non-orthogonal

and 0T2) is designed to replace it in this case.

l. NOTATIONS AND HISTORY

Let (IyB,P) be a probability measure space for each
1: E 9 = {FO,F1,...,Fm}, d be an action space, L be a loss
function which is defined on I X«9 X47 to the non-negative reals

with value variously expressed
<1) L<x.Fi.a> = L<x,F,><a> = xLien.

We assume that for each i, V Li(a) has finite lower integral
a
with respect to F1,

(2) c1 = i: Li(a)dFi < co.

Since the Space (7 serves only as a parameter Space for the
class :5 = {L(a)la é a} of loss functions on I, X 0, it is without
loss of generality to assume that 4’ contains no duplicates in this
sense. To avoid the notational buildup attendant on the introduction
of randomization at this and higher levels, we assume that (7 is its
own extension to the class of all probability measures on the o-field
of subsets of a generated by {L(x,>P)|x E I, P E 9} : given any

such probability measure 5,

<3) a a5 64 a Mag) =J‘ L<a>d§<a>. Vx.P-

a is unique by the assumption of no duplicates. We observe that

E

for each x and P, L is linear in a. For a

t t a1 + (l-t)a0

is in (7 if a and a are, and

O 1
2

(4) L(x,P,at) =‘f L(x,P,-)dat = t L(x,P,a1) + (l-t)L(x,P,ao),

where the first equality follows from (3) and the second one follows
from linearity of integrals.

Hereafter we Shall write integrals in operator notation, e.g.
the integral in (3) would be expressed §[L(a)] or, and preferably,
§[L].

Let .D be the family of all functions d on I. to a such
that Lie d, the function which maps x to xLi(d(x)), isia-measurable
for each i. For d 6.3 we define the risk of d at Fi as the

integral of Li° d with respect to Fi’
(5) R(Fi’d) = Fi[Li° d] s ci'

If G is a distribution on {0,...,m} we define the Bayes risk

against G by

(6) ¢(G) = A G[F1[Lio d]].
.8

We refer to the decision problem described above as the
component problem. ‘When N decision problems each with above
generic structure are considered simultaneously, the resulting
N-fold global problem is called a compound decision problem with
finite state components.

Specifically, let :5 e r”, (3,553) = am“, 3 e g =9“,

6 d = 4N and let .0 be the family of all functions

a
S = (d1,...,dN) from I, to g such that
(7) x Lio da(§)

oz

is 48-measurable for all a and i, Letting

N

(3) W(X.P.a) = N E L(Xa.Pd.aa).

we define the risk of d E .D at P E 9, by

(9) R<P.d) = ftw<§.§,g(§)>1 s u'1 z Nici.

” " i

g E 9* is called a simple (sometimes, simple symmetric)
procedure if dabi) = d(xa) for all 01, a.e. e for some d E .8.
Let § be the class of all simple procedures and let S E S be
denoted by dN. It will follow directly from the definition of 6 in
Section 2 that §C 9:, the subclass of .2 equivariant under the
permutation group. AS functions of 3, A R(P,S) and A R(P,S)
will be called the Simple envelope and the equivariant envelope,
reSpectively. It is well known (cf. (27) ff.) that the former coin-
cides with the component Bayes risk ¢(N) with N denoting the
empiric distribution of P1,...,Ph.

The compound decision problem was introduced by Robbins (1951).
He argued that a bootstrap procedure which first estimates the
empiric distribution of P1,...,Pfi and then plays Bayes against
the estimate within each component may have its compound risk uni-
formly close to the simple envelope.

Hannan and Robbins (1955) considered 2 X 2 9 X a and
(Theorem 3) bounded the average loss of a bootstrap procedure by
the sum of an error of estimation and a loss-weighted Glivenko-
Cantelli measure of deviations of the empiric distribution of

x1,...,xN, thus obtaining strong convergence to zero uniformly

in P* of the difference of the average loss from the Simple

envelope for all correspondingly good estimators. Risk convergence
(Theorem 4) followed as a corollary. Oaten (1969) permits loss
dependence on x, replaces Bayes by any of a wide class of e-Bayes
and otherwise generalizes these results to m X n 9 X 4 (Theorem
1 and its Corollary) and to certain compact ‘9 X compact a’ with

L continuous for each x (Theorems 4 and 5). Under continuity
and other restrictions on densities, analogues of generalizations
to m X n 9 X a were given earlier by Suzuki (1966a).

Hannan and Robbins (1955) also introduced the class of
equivariant procedures and showed (Theorem 5) that the difference
between the simple and equivariant envelopes convergasto zero
uniformly in P as N t m. The proof depended heavily on a measure
theoretic lemma Specializing Theorem 11.1 of Hannan (1953). Our
Theorems 1 and 2 (CTl) and (T2) of our Summary) are a strengthened
generalization of their result, with Theorem 1 correspondingly re-
lated to Theorem 3 of Hannan and Huang (1969b) and Theorem 2 follow-
ing as a somewhat involved corollary to Theorem 1.

Hannan and Van Ryzin (1965), for 2 X 2 G’X GK and Van Ryzin
(1966), for m X n 9 X a, have established a rate of OCN-k) (and
under additional restrictions on '6’ and L, OGN-1)) for uniform
risk convergence of bootstrap procedures based on estimators which

are averages over x of a suitable kernel.

1,...,xN
The importance of our results stems from the basic character

of equivariant procedures in the compound problem. Until Oaten's

(1969) e-Bayes relaxation, all of the bootstrap procedures con-

sidered were essentially equivalent (cf. Lemma 3 of Oaten (1969))

to equivariant procedures. The equivariant envelope is then a

clearly more appropriate yardstick of performance than the Simple
one. The results themselves have already been used by Oaten (1969),
together with his afore-mentioned Theorem 1, to prove risk con-
vergence for a wide class of equivariant uniformly-e-Bayes pro-
cedures (Theorem 2).

AS noted in Section 3 of Hannan and Huang (1969b), a gen-
eralization of the underlying measure theoretic lemma, Theorem 2
of Horn (1968), turns out to be distinctly improved by an immediate
extension of the afore-mentioned Theorem II.1. Her corresponding
result on the difference between the envelopes (Theorem 1) inherits
the deficiencies of her Theorem 2 and only Shows convergence to
zero for each ‘3 with a Stronger restriction on «9 than pairwise
non-orthogonality, with a7 finite and with L constant with
reSpect to x.

Considerable other work relates to equivariant procedures
in the compound problem. Stein (1956) and James and Stein (1961)
(cf. Stein (1966), where the heuristic of the procedure is revealed,
and Cogburn (1965)) obtained strong results with a Gaussian Squared-
deviation-loss estimation component problem. In an important general
development, Cogburn (196?) imbeds the compound and Empirical Bayes
problems in a general theory of stringency. Section 4 of Samuel
(1967) (cf. Robbins (1962) and Suzuki (1966b)) investigates a pro-
cedure Bayes against uniform prior on proportions for the simplest

2 X 2 example.

2. EQUIVARIANT DECISION PROCEDURES AND
SYMMETRIFICATIONS IN A COMPOUND DECISION PROBLEM

Let .8 be the permutation group on N objects. The generic
element g 6.8 will also be used to denote the transformation induced

by g:
10 = ,..., .
( ) g X (yg1 ygN)

Letting g(B) = {g x| x E B} for B 6.6, it follows from the trans-
formation theorem (Theorem 39.C, Halmos (1950)) that for each P 6.9

and g E .9, g P is in 9 and satisfies

~

(11) P03)

N”

(g§)(g§). g 6 [3.

Furthermore, it follows from the definition of W in (8) that, for

each a E 4, ga is in d and

(12) W(§:B:S) = w(g?ssg£:gg)-

Thus the compound decision problem is invariant under .8.

d 6.3 is equivariant (under .g) if for all g E.&,
(13) d(gx) = gd(x) a.e. a”.

Hannan and Robbins (1955) and Ferguson (1967) use the term invariant
procedures instead of equivariant procedures. The latter was
suggested by Wijsman (1968) to describe functions which transform
properly rather than "invariantly". In our further references we

will presuppose this change has been made.

7

It follows directly from definition (13) that d 6 6 if and
N-l
only if there exists a function v on I X I to <7, symmetric

on IN-l, and such that for all a

(M) dgy=vaw$> améfi
where
(15) Ed = (x1,...,xa_l, xa+1,...,xN).

Equivalently, d 6 6 if and only if there exists a function 6 on

I X IF to <7, symmetric on IE, such that for all a,
(16) d (x) = 6(x ,x) a.e. 6N.

Q ~ Q ~
It follows from the definition of S and 6 that
(17) S s: 6.

For each d 6.3 and g 6.9 define dg, the g-conjugate of

d, by

(18) gg(§) = g’1[g(g§>]-

8

Thus d 6 6 if and only if d = d for all g. Define the

*
symmetrification d of d 6.0 as the average of its conjugates,

N

(19) 9* = 010'1 2 dg.

It follows immediately that

III
m
l a.
___x-
i a.
m
It»
bur-J

*
W 5? = 2

Corresponding results hold for Subgroups of the permutation
group and relate to certain extended (cf. Swain (1965), Johns (1967),
Gilliland and Hannan (1969) where only the sequence version is con-

sidered) compound decision problems.

3. REPRESENTATION OF EQUIVARIANT RISK

As a basis for comparing the Simple and equivariant envelopes,
we obtain in this section a convenient representation of the risk
function of equivariant procedures and relate to the Bayes risk
against a certain uniform prior.

For each d 6.9’ and g 6.8, it follows directly from de-
finition (9), (12) and the transformation theorem that the risk oftfiu!

g-conjugate is

) = EEW(x.P.dg(§>)]

(21) = fCW(s§.sf.g(g§))]

Averaging the above over .& gives the risk of the symmetrification

*
d .

<22) R<§,g*> = (N!)'1 2 R<s§.g).
8

Since d = dg for equivariant d, (21) also implies

~

(23) R(P,d) = R(gP,d), d E 6.

Letting Ni(E) = #{a‘P& = F1}, and letting N(P) = (NO(P),...,Nm(P))
be a convenient index of the P-orbit, we shall hereafter write
R(N,d) for LHS (23) with N = N(P), and denote the equivariant

envelOpe by

10

(24) 1'93) = Rq.g>.

396)

Hannan and Robbins (1955) consider the class E’ of all

~

.8 satisfying the constant risk prOperty (23). They Show that

d E
$(N) coincides with A RON,d), the "risk-invariant" envelope. We
~ E N"

N

wish to remark that the class 6’ need not be considered separately
*

because, for each d 6 £3 its symmetrification d has the same

risk according to (22).

For 6 6.9 it follows from the definition of risk (9) that
NR(E:§) = g E[L(XQ,P&) o 5a(§)]

=2 2 (Fixfa)[1*i°5o,3’

1 a‘ Pa=Fi
where Fi acts on xa and 3a E (P1,...,P&_1, Pa+1,...,PN) acts
V
on x .
~a
In particular, if 6 6 6, then by (14) 6a (and therefore
Li<>6a) is symmetric in fa’ and, with Nji E Nj-l or Nj depend-

ing on j - i or 1 # i and 2 denoting Sum over i such that

Ni > 0, we have the following representation of equivariant risk,

N..
J1
= x
(25) NR(N~,§) >3 NiFi j Fj [Lia 61].
Nji v
where Fi acts on x1 and X Fj acts on x1. The order of the
N J "
Fj in X Fjji is immaterial Since the integrand is symmetric.
J
Let p be any measure dominating ‘9 and let fi = dFi/du.

[L10 61] by T(61) for g 6 6, (25)

Abbreviating 2 Nifi g Fj

is expressible as

(26) mm”) = u[T(61)]-

12

If 6 E 3, say 6 = dN, then 61 is a function of x1 alone.

Thus

2

_ _.1_ o
(27) R(N,d ) - E N FiELi d].

N

The infimum over .8 of LHS (27) is, by definition, the Simple
envelope. The infimum over .8 of RHS (27) is, by (6), (raj/N),
the Bayes risk against the prior N/N. (This is also the infimum
over .8 of the (N/N)N-weighted risk but we shall make no use of
this interpretation). Hereafter we abbreviate WON/N) by (3(5).
We now Show that WCN) is the Bayes risk in the compound
problem against the uniform prior on the orbit indexed by E.
Let UN denote such a prior. Applying the transformation theorem
to the~mapping g _. g3, we obtain from (22) that

(28) R634”) = 2 R<g.g)uN(g>.

N

N

20

The infimum over .8 of the LHS above is $01) by (20). The

N

infimum of the RHS is, by definition, the Bayes risk against UN’

N

say, R(UN) . Thus

~

(29) ME) = MUN).
and therefore d 6 .8 is e-Bayes if and only if

(30) 1101,?) s 1793) =+ e.

4. THE DIFFERENCE BETWEEN.THE TWO ENVELOPES

WHEN 9 IS PAIRWISE NON-ORTHOGONAL

In this and the following section we bound the difference
between the Simple and equivariant envelopes. Since
v |F(B)-F
36/3 1 j 3

only when ‘9 is pairwise non-orthogonal.

(B)‘ = 1 when Fi I.F , Theorem 1 is of interest

Theorem 1. 'LEE pij = V ‘Fi(B) - Fj(B)‘, ci satisfy (2),

B
K(p> =fi-(jig-)3”)(1-p)'3/2 flilfiE p = v pij- Then
1.)
... 2 _
(31) mg) - Mg) s {2K<p) 2 cifin 6.
i

nggf. For each N and each equivariant 6, we will con-
struct a simple procedure dN whose risk at N is close to the
risk of E at E. To bound the difference in risks we use Theorem
3 of Hannan and Huang (1969b), renotated here for our application

by the use of relation (14) of that paper:
For any positive integer N and any non-negative integral

partitions N and N' of N,

XFNi[ x1?1 0 =*512

Vii icpl-i item 549 P 1
(32)

snl<();;/\'1(‘N'-N)2

p i 1 i i ’

. = ' - = ' 0
With n #{klNk f NR} 1, Ai (Ni A Ni) + l for all i,

and p =V{pij‘Ni #N;_. N #le-

J
For 6 E 6 consider R(N,6) in the form (25). For given i
and x , let

1 13

14

L,o6

_ _ . 1
(33) m the x1 section of V—EEIET .
a

From (2), 6 takes values in [0,1] and, from (14), m is symmetric
in £1. Applying (32) to the integrand with reSpect to F, in
~ 1

RHS (25) for each i yields

(34) x FNji[L o 6 )2 x FNjJEL o 5 ] - {v L (a)}{1<( )-1—+-1—)}%
ijiljjilai piJ(NiNJ’

for any J 6 {O,...,m}. With J such that NJ 3 V Nj’ we weaken

J

the bound (34) by simultaneously replacing ér' by fir' and K(piJ)
J i

by K(p). Taking upper integrals with respect to F1 and weighting

by Ni’ we thus obtain

N
iJ _ a «E
(35) NR(N,§) 2 )3 NiFi )j< Fj [Lie 61] (2K(p)) ziNici.

with C1 given by (2).
We now construct the simple procedure dN. Since for each
x and i, x Lio 6(x1, ) is a symmetric function of Ԥ1, it

1
1
follows from the transformation theorem that

N
N . = .
(36) x Fj Lx Li° 61(x1. )1 fix Li( )1.
J 1 1
N J -1
with g = (X Fjj )[61(x1,-)] . We note that g depends on x1
3
but not on 1. By assumption (3), there exists ax E<7 such that
l
RHS (36) = L,(a ), V'i. Letting d be the function mapping
x1 1 x1
x to such a , we see that, for each x ,
1 x1 1
(37) x Lie d(x1) = x Li(ax ) = LHS (36),

l l 1

15

which is B-measurable. Therefore d 6 .8, and dN is simple. Also
by (37) we recognize the first term of RHS (35) as N RHS(27)

which is bounded below by N N(N). Thus
35 %
(38) NRq.g) 2 N MN) - <21<<p)) z Nici.
N i
Applying the Schwarz inequality to the Sum on RHS (38) yields
- 2
(39) R0345) >- wcg) - <2K<p))’in ’56: c1)?
i

Since (39) holds for all 6 E 6, this completes the proof of Theorem
1.

5. THE DIFFERENCE BETWEEN THE TWO ENVELOPES WHEN
«9 MAY HAVE SOME PAIRWISE ORTHOGONALITY

In this section we derive, essentially as a corollary to
Theorem 1, a useful bound for the difference when ‘9 may have

some pairwise orthogonality.

P = V{pij|pij < 1}. Then
(40) (MN) - MN) s 2m{21<(p) Z‘. ciffu't.
i

2522:. The plan of our proof iS first to decompose the whole
problem into pieces of sub-problems, each satisfying the pairwise
non-orthogonality condition. For arbitrary 6 E 6 and a Special
choice of d to be (2tn+1 - 1)e r-Bayes with respect to N, we
represent the difference in risks as the sum of differences of Simple
and equivariant risks in the Sub-problems with the Simple procedure
being e-Bayes against the restriction of N to the sub-problems.

For each Is;{0,...,m} let 5={Fi\i61} and N= EN.

1
161
The subproblem determined by 'éfi will be called the I problem.

Let .8, 6, N, Ufi’ T, W, R, T, and ; denote the I problem counter-
part of these symbols without the delete Sign v . For simplicity
~/

V
we omit the delete Sign on * and R hereafter.

Let

(41) II = {x e I‘fi(x) > 0 iff i e 1},

l6

17

and let ”I be the restriction of u to Ii. Since I = 2 II
I

it follows that p, = E ”I.
I
Let 5 6 6. The risk of 6 is of form (25), which is

expressible as integrals with respect to ”1’

N . N
(42) NR<N.5) = 2 uI{ z: Nifi x FJ.jl x F.j[Lio 61]},
" " I i6I jEI jéi J

N
h e tak X j v ... f h . F
w ere w e jéIF to act on (xN+l’ ,XN) or eac I or
each I and each (x1,...,xfi) in RHS (42), there exists, for

the same reason behind (36), a distribution g over (7 such that

N
J ._. .
(43) jéIFj [L10 51] SILi], 1 6 I.

By assumption (3) there exists ag E d’ With Li(a§) = §[Li] for

all i 6 I. Letting Kl be the function mapping (X1,...,Xfi) to

such ag we see that L e K gives RHS (43) and is thus 6F-

1 1
measurable for each i E I. Furthermore, 6 6 6 implies the symmetry

of El in (x2,...,xfi), and therefore 3 is the first component

1
of some 6 E 6 constructable by the use of (14). Thus (42) yields

the representation,

~~

(44) NR(N,6) = E HI[T(EI)]-
I

For each I, let dI be e-Bayes in .8 against N/N and
let d = z IIdI where II serves as the indicator function of
I

itself. We note that d 6.8 and d = dI a.e. uI. Thus by (27),

by u ==2 “I and by the fact that Lie d = Lia dI a.e. ”I and IS

constant with reSpect to (x2,...,xfi), it follows that

(as) Nags“) = z ultiwln.
I

18

The difference between (45) and (44) is

z uI[T(dI) - T<61>]-

(46) Nags”) - 1293.9}
I

v v v V
For each I, define h = (h1,...,hfi) 6.8 by

N

6v _ ..f
Oa(X-1-,-...,X1\q’) 1 X0! E II

(47) ha(X1,...Xfi) =€

 

d (x ) otherwise .
I a

L

V V
By (14) we see that h 6 6. By direct calculation using (26) and

v
the definition of h,

v v V
NRqu = sltmln + (u-sI)[T(dI)]
(48> v 6 fi v v v
= NRq,dI) - uI[T(dI) - T(61)].
Since dI is e-Bayes with respect to fi/N and h 6 6; (48) yields
(49) ulticdl) - 161)] s MT) - 1763)) +Ne-

It follows from (38) that, with 5'= V{pij‘i,j 6 I},

(50) RHS (49) s (21(6))15 2 Nfci +-Ns.

iEI
Summing (50) over all I with “I f 0, we obtain an upper bound
for RHS (46). Since n1 # 0 implies E < l we shall weaken (50)
by replacing 5 by p, and then dropping the restriction on the

summand. Thus

% l

(51) R(N,dN) - R(N,6) S (2K(p));5N-1 2 E Nici +-N-

V
€2No
I 161 I

Since (51) holds for all 6 E 6 and all e > 0, and therfore for

19

e = O, the proof is complete upon using the Schwarz inequality in

(51):

(52) z: 2: NEC. =2 2N.cisz N (26:)

PART b
A STABILITY OF SYMMETRIFICATIONS 0F PRODUCT MEASURES WITH
FEW DISTINCT FACTORS

O . SUMMARY

Let .9 = {F ..,Fm} be a class of probability measures on

0"
*
(1,8). For any signed measure T on 8“, let T be the average

of T3 over all N! permutations g and let “T“ = V{‘7(C)‘:C 6 5”}.

_ _ 12. 2 3/2 -3/2
Let dij HFi FjH and K(x) 11 (5) x(l-x) . For any non-
negative integral partitions N = (N ,...,N ) and N' = ',...,N')
... O m ... 0 m
= ' .. = t .
of N, 1:et 61 N'N1 Ni and Ai (Ni A Ni) + l. Wlth
. . *
T = X Fi1 - X F11 and n = #{iléi * 0} - 1, we bound “T H2 by

(T3) n K(d)Z a: A;1 with d = vfdijlai # 0, bj # 0}

and, if .9 is internally connected by chains with non-orthogonal

successive elements, by
(T4) 1 m K(d)(2‘6 ‘)2(Z A-l) with d = V{d ‘F F }
2 i i ij i/1 j ‘

The bound (TB) is finite iff the F1 are pairwise non-orthogonal

and (T4) is designed to replace it otherwise.

20

1. INTRODUCTION

Section 2 investigates some general properties of signed
measures and their symmetrifications w.r.t. general groups.

Section 3 specializes to the permutation group and notes a contrac-
tion effect of probability factors in product signed measures. The
properties developed in Sections 2 and 3 will be used throughout
the paper and, in particular, in Lemma 2, in the completion of the
proof of Theorem 1, and in the proofs of Theorems 2, 3 and 4.

Section 4 proves Theorem 1, which is the special case of
(T3) for m = l and 61 = 1. This is the main result of the paper.
Its proof contains a detailed outline of itself including Lemmas
2, 3 and 4. An example, consisting of the simplest special case,
shows that the bound of Theorem 1 is sometimes asymptotically
sharp to within a factor of 3.18... .

Section 5 proves Theorems 2, 3, and 4, all as corollaries to
Theorem 1. Theorem 2 is the Special case of (T3) for m = l,
Theorem3 is (T3) and Theorem 4 is (T4).

Our main results are a strengthened generalization of
Theorem 11.1 of Hannan (1953). The latter is easily characterized

in terms of the m = 1 case of (TB),
* 2 2
(T2) Hi H s K(d01) 61 (A0 A ) .

amounting to the assertion that, for m = l and fixed F0 er1’

21

22
(T 11.1) HT*H2 -» o as RHS(TZ) —. 0.

As partially indicated in Section 3 and Section 5, the derivation
in Section 5 of (T3) and (T4) as corollaries of Theorem 2 would
equally well yield weakened (as (T 11.1) weakens (T2)) forms of
(T3) and (T4) as corollaries of (T 11.1).

A corollary of the 61 = 1 case of (T 11.1), the Lemma of
Hannan and Robbins (1955), was there used to show (Theorem 5) that
the difference between the simple and equivariant envelopes con-
verges to zero. A generalization of the Lemma, Theorem 2 of Horn
(1968), is shown in Section 3 to be improved by the 61 = 1 case
of a rather immediate corollary to (T 11.1). The Special case of
(T3) with two non-zero 6i is used in Hannan and Huang (1969a)
(Theorem 1) to bound a more general case of the difference. A

similar application, Theorem 1 of Horn (l968),inherits the

deficiencies of her Theorem 2.

2. PROPERTIES OF SYMMETRIFICATION AND OF AN ii-NORM

Although in this paper we will only be concerned with the
difference between two symmetrified probability measures, some of
the prOperties used in our proofs hold true and are easier to prove
in a more general context. This section investigates some of these
properties of symmetrified signed measures and their ifi-norms.

Let (Q,CD be a measurable space and .3 be a finite group
of measurable transformations g on (u,CD. For a signed measure
T on (u,CD, we define Tg as the induced signed measure and T*

as the symmetrification of T by
*
(Tg)c = T(g(c)) c E G , T = AV(Tg)

where AV denotes the average over g 6.3. Thus symmetrification
(*, hereafter) is a linear Operator.
*
For any real valued function f on u, define fog and f

by
<fog)y = f<gy> . f* = Av<fog).

* *
T and f are said to be symmetric if T = T and f = f , res-
* *
pectively. Since T g E T , T is symmetric iff T E Tg.
Throughout this paper we shall denote supremum and infimum

by V and A, and express integrals as left Operators by
w<f> = j f<y>d¢<y>.

23

24

For any signed measure r, define an eel-norm of r by
(1) Hr” = V{\T(C)| :c e a}.
It follows from the Jordan decomposition that
<2) M -—- M v urn
and hence, if T(l) = 0,
m M = “in = urn.
In particular, if p and Q are probability measures, then
(4) o 5 HP - Q“ s 1,

with equality at 0 iff P = Q, and equality at 1 iff P i_Q.
For use in the proof of Lemma 1, let u be a measure

3' CIT/d“ ”183° Since dT+/du = (dT/du)+ and dT-Idu = (dT/dH)-a
“TH = MdT/du)+ v u(dT/du)'.

Hence, if T(l) = 0,

<5> 2M = lefi-l).

It follows from the transformation theorem (Theorem 39.C, Halmos

(1950)) that

it =£1a
(dpfig dug’ 863-

*
In particular, if u = u , AV of the above equality yields

*
dT * _ dT

(6) (31:) - d“. '

25

Let u be a measure and let h and f be such that the
* * *
products h f and hf are u ~integrab1e. By the transformation
* * * *
theorem and symmetry, u (h ng) = u (h f) for all g 6.&. Averag-

ing over .9 and interchanging h and f yields
* * * * * * *
(7) u-(hf)=uChf)=u(hf).

*

For the Special case of (7) obtained by letting h = du/du , we have
* *

h = l by (6) and therefore, if f (and hence also f ) is

*
u -integrable, then
* * * *
(8) N(f)=u(f)=u(f)o

For use in the completion of proof of Theorem 1, we note that,

by subadditivity of norm and by HTgH E “T“ for a signed measure T,

(9) Wu s Av \bsu = urn.

It follows from norm subadditivity and the Schwarz inequality (here-

after referred to as NS-SI) that if T1 are signed measures then
2 2
(10> us: mm s o: n: M .

If T = X T is a product signed measure, HT“ is simply

T12
related to the correSponding norms of its factors. We abbreviate
by omission subscripts on the norms. Since
+++ --__+- -+-
T - (T1 x T2) +071 x ¢2), T - (T1 X T2) +-(vr1 X T2), it follows

easily that

(11) HTIH'HTZH 5 Ha X T2“ 5 2HT1\\°HT2H,

with the first equality iff either T1 or T2 is a measure or the

26

negative of a measure, the second equality iff T1(1) = T2(1) - O.

In particular, if T is a probability measure, the first equality

2
of (11) yields

(12) \hl X 12“ = “11H.

To assist in comparing our results with Theorem 11.1 of

Hannan (1953), consider a signed measure T with T(l) = 0. By

(3).
(B) M = Ml = V{T(f)|0 s f s 1}.

*
Since T (l) = 0 by linearity of * operator, upon applying (13)

to T*, and applying (8) to T*(f), it follows that
* * *
(14) Mr H = v{r(£ )|o s f s 1} = v{r(£)|o s f = f s 1},

*
with the second equality following from {f ‘0 s f s l} =

{f|o s f = f* s l}.

3. PERMUTATIONS; CONTRACTION EFFECT OF PROBABILITY FACTORS

Henceforth we Specialize .3 to be the group of transforma-
tions on (Ig3)N induced by the group of permutations on N
objects, where (IJB) is a measurable Space. We also let .&
denote the permutation group itself. Thus a generic element
3 6.9 will be used both as a permutation and the transformation
gx = (xgl’°'°’ng)'

The following lemma will be used in the successive extensions,
in Section 5, from Theorem 2 to Theorem 3 to Theorem 4. Starting
from Theorem 11.1 of Hannan (1953) rather than Theorem 2, Lemma 1
together with NS-SI (10) would yield an extension paralleling our
Theorem 3 extension from Theorem 2. It will be Shown that Lemma 1
alone would yield an extension improving Theorem 2 of Horn (1968),
where there is a stronger restriction on the F1 than pairwise
non-orthogonality, and where the non-zero 6i are 1 and -1
respectively.

Lemma 1. ‘If T = T X P for a signed measure ‘T with

 

T(l) = 0 and a probability measure P, then, abbreviating affixes

 

 

on * 229.22 “ H by omission,
\\«r*\\ s \\<>r’>*u-

* v *
Proof. Since r(1) = 0, “T H = V{T(P(f))|0 s f = f s 1}
by (14). Since fi= P(f) is symmetric in the remaining variables
and since 0 s f s 1, one more application of (14) completes the

proof.
- 27

28

N. n!

For our extension of Theorem 11.1, let T = X F1l - X Fi1
where the F1 are pairwise non-orthogonal and fixed, and the
6i =‘N; - Ni are zero except for two i's. By judicious choice

v Ni
of g, we have Tg = T X P with P = X{Fi | 61= 0}. By
* * * v *

(Tg) a r and Lemma 1, Hr H 3 1| (r) H which, by (14) and Theorem

II.l, converges to zero as this case of the bound in (T2) does.

4. TWO DISTINCT FACTORS WITH UNIT DIFFERENCES IN MULTIPLICITIES

Theorem 1. Let F and F0 be pfmeasures and let

 

 

1
N = (N1,N0) be an integral partition of_ N with N1 2 0 < No.
-3 2

With d = “Fl - F0“, with C(d) = d(l-d) I ang_with

2 -1 -1 -l
K(d) = (.4) (6 +'5 ) C(d)/C(.4)

N N N +1 N -1

l o * 1 o * 2 1 .1_

(15) “(F1 x F0 ) - (F1 x F0 ) H s K(d)(N—_1+1 +NO).

2522;. Since both sides of (15) vanish if d = 0, hence-
forth assume F1 # F0.

The proof proceeds according to the following outline: As
in Hannan (1953), a parametric family of densities of the F1 is
introduced and some of their moment properties are related to d.
Starting as in Hannan (1953) but then weakening by the Schwarz.
inequality, Lemma 2 obtains a family of upper bounds for a slight
generalization of LHS (15). Lemma 3 develops lower bounds for
modal binomial probabilities for application to the denominators
of a bound of Lemma 2. Lemma 4 uses characteristic function in-
versions to bound a generalization of the numerators of the bound
of Lemma 2. The bound of (15) is then obtained as the minimum of

d2, from Lemma 1, and a bound resulting from the application of

the other lemmas.

For 0 < p = l-q < l and i 1,0, let

(l6) Fp = pF1 + qFO , fi = dFi/de,

29

30

(17) e = F1(f0)’ P = p F1<fi> = 1 - Qi(= 1 - q Fo(fi))’

i

and note
(18) F.<f.> = F (£2) > (r (f >)2 = 1
1 1 p i p i

by the Schwarz inequality, with equality eliminated by Fp # Fi.

Thus
(19) e = p'la - q F0050» < 1.

Let u be any o-finite measure dominating the Fi’ let h1 = dFi/du

-l
= . ' = h h '
and let hp ph1 +qhO Since flfohp h1 0 p 2 h1 A ho, it
follows that
(20) e 2 u(h1 A ho) = l - d.

Finally, note that from (17) and (18)

(21) P1 - P0 = 1 - 9, P521 = pq F1(fi)FO(fi) > pq e.

For integer N and p 6 [0,1]N, let b(k;p) denote the
generalized binomial probability of k successes in N independent
trials with success probabilities B.

At the cost of notational complications, the following lemma
could be stated and proved for m instead of 2 probability
measures.

Lemma 2. For non-negative integral partitions of. N,

N1 N0 N1 N6

= '= " . = I:
N 0N0,N1) and N 2 (NO,N1), define E F1 X F0 ,2 F X F

and T=F-F'. For O<p<1

31

 

 

 

N' N‘ N N
(dF'* =;b(N1 P11 P00) b(Ni;P11 P00)
(22) = .
(dFN bCN1;PN ) bCNisz)
b(1<;1>r PN’r) k=N' r=N'
(23) 4\\T*\\2 < N o J l J l
b(k;p ) k=N1 r=N1

* * * *
Proof. Since F (dg' /dF:) = F' (dF /ng), the second
equality in (22) will follow from the first.
*
By (6) the integrand in LHS (22) is (dF'ldFE) , whence, by

(8), LHS (22) is AV of

dF'
(24> F('=§‘°g)
" dF
P v
. . . . N1 f
Since the integrand in (24) is the product H1 f1(xga)IlE.+1 0(xga)

of F-independent variables, the integral is expressible, in terms

of K = #[cr >N,'1‘g01 < N1], has the product

N'-K N -N'+K N -K
0
[F0<fo>1 ° [F1(f0)]K[Fo(f1)] ° [F1(fl)] 1
(25) -N -N KN'-KN -KN;-N 'I'K

_ 1 0-0 1 .11 =
-p q PoQo P1 Q1 H00-

Then (22) follows since

 

NI NI

N' N' N 1204131 PO 0)
(26) AV H(K(g)) =2(k0)(N1.1_1k)(N ) 111(k) = N
bCN1;P)

with the first equality following from the transfermation theorem.

From (5) and the Schwarz inequality,

(27) 4\\T*\\2=1:(:r|—N’\)]2<:(|:J—;2\ ) = ”(31;)-
Pp Fp dF

Four-fold application of (22) to this bound results in (23), completing

32

the proof.

For integers 0 s k s N, let

(28) M = {01-1)qu b(k;pN) with p = (Mn/(NH).

= -1
Lemma} aNN-l-k awkzaNOSe fig N100.

Proof. Note that f(x) = (l +-x1)x+%§ as O < x t m

 

since, with t = (2x + l)-1, log f(x) = l + t 2/3 e+~t4l5 + ...

The lemma then follows from the representations,

= ELL = km; N'”
<29) am. ENk-l f(N-k) aho H f(N- j) ’ aNo= {Ii—+1} tm)

The following lemma uses characteristic function inversions
to bound a mixed second divided difference of generalized binomial
probabilities. In our application to certain numerators of (23),
the generalized binomials involve only two distinct probabilities
but the added generality simplifies the proof and may serve to
motivate other applications.

Lemma 4, For integers o s k s M, P = (p1,...,gfi) e [0,1]M

N

 

2
and (P,P+h) 6 [0,1] less the diagonal, let A and A reSpectively
denote the divided difference operators from k to k+l and from

P £2. P+h. ‘With B(x)= (4/n)on W(l y) %e 2x xydy and

2 M
0' - 21 PCY(1-Pa),

(30) \c b b<k;(P.§)>\ s 13(02).

x3/2 -a

(31) B(X) -* (211) as X 1 °°,

3/2

(32) s = sup {x B(x)\o s x < o} = .545447...

33

Proof. Note for later use that, with ¢(u) = Pe1U +-Q the

ch.f. of a Bernoulli variable with parameter P = l - Q,

(33) ‘¢(u)‘2 = l - 4PQ sin2 2 5 exp - 4PQ sin2 2 .
2 2
Since i_\.e-ink A ¢(u) = e.iUR 4 sin2 g, the iterated divided difference

in (30) is given by the Fourier inversion

l -iuk , 2 p
(34) fig e 4 sin 2

2n <50, (u)du.

H213

By application of (33) to the Qa’ the modulus of (34) is bounded

by
2 2n
3 . 2 p -20 sin _ 2
(35) n I3 Sin 2 e Y'du - B(o ).

(Since v-lsin v I, on 0 < v 3 11/2, RHS (33) 5 exp - 4l?Q(u/Tr)2 on

‘u‘ s n. The correSponding weakening of (35) implies

5 -
s < n /22 7/2 = 1.553 ...
2 2 5/2
.2 - 2 2 -2x(u/n) _1, 2 -2x(u/Tr) _ 11 -3/2
B(x) S n f3 Sin 2 e du < n_f: u e du - E773'x .)
With IO, I1 denoting the modified Bessel functions,
2 l -% -% -2x

B(x) =[-;}“Oy <1-y> e ydyl'

(36)

= [-Ze'x10(x)]' = 2e'x[10(x) - 11(X)].

where the second equality follows by differentiation from the

correSponding Laplace transform (cf. 29.3.124 of NBS-AMASS (1964)),

' =
and the last from I0 11.

In view of (36), (31) is an immediate consequence of the usual

asymptotic expansions of IO and 11 (cf. 9.7.1 ibid).

34

%

3 2 -
To verify (32), first note that [x / B(x)]' = x e xF(x)

with
F = (3-4x)I0 + (4x-l)Il.

Since F(l) > 0 > F(2), the behavior of F' will imply that
3 x0 6 (1,2) with F s 0 on [0,x0] and F < 0 on (x0,w) as
follows. Since 11 s x10, xF' = x(4x-5)IO +(l-x)(4x+l)I1 S -(15/16)xIO

on [0,1]. Since x(4x-1)F' = (4x+l)(1-x)F - 3I l s x and

O,
F(x) 2 0 = F'(x) < 0 and hence I s x0 and F(xo) = 0 = F < 0 on
(X0 ,m) °

Since, in addition,.00012>’F(1.452) >’0 > F(l.453) (p. 228

BAAS v VI (1937)),

a -1.452

/2B(l.452) 5 (1.452) e (.00012)(.001) < 4(10'8)

0 S s - (1.452)3

which results in (32) and thus completes the proof.

Completion g pro_of_ pf Iheoreml. From Lemma 2 with Ni = N1+l
we have this case of the bounds (23). Let p = (N1+l)/(N+l) hence-
forth. This choice insures both denominators in (this case of) (23)
are equal and, by Lemma 3, are bounded below by {(N-l)pq}-%aNo.

Application of Lemma 4 to the mixed second difference remaining in

the numerator results in the upper bound s‘h‘o-3, with

2
h = Pl-PO = 1-9 s d and o = NlPlQl + (NO-npooO > (N-l)pq(l-d)

/ /2f

by (20) and (21), so that, with cN 4'1s(1~i+1)3 ZCN-1)'3

(N);
4-lse = .3706 ... as N 1 w,

1 l
c C.(d) — + ——).
N (NIH No

(37) \\r*\\2 a 42m c<d>{<N-1>pq}'1

 

35

*
From Lemma 1, “T “2 s d2 independently of N. Since pq
is maximal at p = %, (37) is minimal w.r.t. N given N at

N1 = [N/Z] and, therefore, exceeds d2 for all d iff

(38) max d2/C(d) = (.4>2/c<.4> s aN([‘i:—21'1 + Eli-1]“).
d

Since RHS (38) 1 w.r.t. N with first violation of (38) at N = 10,

(N of (37) is replaceable by c9 = .5185 ... . This in turn can

be improved by first increasing 8 to insure equality at N = 10,
thus replacing cN by (.4)2(o'1 + 5'1)'1/c(.4) = .5070 ... and
completing the proof.

That the bound of the theorem is sometimes relatively sharp,
will follow from examination of the simplest Special case.

Example. Let F0 and F1 be Bernoulli probability measures
b( ;§0) and b( ;§1) with 0 = g0 < €11< 1. Then

N N N '1 N
0 * 0
and (F0 X F1

(F0 x F1 ) N

measures on {0,1}N, putting mass, 6:)-1b(k;§11) and

N _1 N +1 N

(k) b(k;§1 ) respectively, on every x in {0,1} with exactly
N " N+l

for b(k;§11) and using b(k;§11 ) =

+1 *

1 ) are symmetric probability

k ones. Writing bk

glbk-l + (1-§1)bk’

*
(39) ZHT H =‘E1bk -§1bk-l ‘ (1-§I)bk‘ = glilbk ’ bk-l‘ = 2glbm

with m the greatest integer not exceeding (N1+l)§1. As

(N1+1)§1(1'§1) *‘ma (N1+1)§1(1'§1)b: ~ (2n)-1 and therefore

N +1

E
* 2 1 1 1
(40) “T H ” 2n l-§1 1 °

36

When gl 4 0 and No/N1 d m (which is the most favorable case for

the following comparison), the bound of the theorem exceeds RHS (40)

/2

only by the factor .8n (30/11)(.6)3 = 3.18...

5. m+l DISTINCT FACTORS

We now obtain various extensions of Theorem 1 as corollaries
to that theorem. Theorems 1, 2 and 3 represent successive extensions
each subsuming, yet corollary to, its predecessor. Theorem 3 is
vacuous unless the F1 are pairwise non-orthogonal, and Theorem 4

is designed to replace it in this case. Thus, as implied in the

summary, our final results are merely Theorems 3 and 4. Let

Ni N;
¢=XFi -XFi ,dij =HFi-FjH
i 1
(41)
0. =N! -N, , A, = (N3 AN.) +1, i,j =0, .,m.
i i i i i 1
Theorem 2. For m = l,
* 2 2 l l
(42) Mr H s K(d01)51(A + A ).
0 1
Proof. Assume without loss of generality 61 2 l (we may
rename N and N' otherwise), and for j = l,...,61, let
N -j+l N +j-l N -j N +3
= 0 l 0 l
(43) Tj F0 x F1 - F0 X F1

Since T = Z Tj, it follows from linearity of * and NS-SI (10)

that
(44) H'r*H2 s a, 2 MHZ-

Applying Theorem 1 to each summand in RHS (44) completes the proof.

37

38

Theorem 3. .222 n = #{il6i # 0} - 1 23 positive (otherwise

T* = 0), 95.19.19}. d = V{dij‘6i # O, 5 f 0}. Then

3

\\r*\\2 s n K(d) z (sf/£1.

Proof. Given N and N' we construct a sequence of partitions

22

= NO’ N1,...,Nr = N', for some r s n, as follows. To construct

1=CN

22

10
and let t be such that 6t has opposite sign with 68. Let

"'°’N1m)’ let s be such that |os| = A{‘6J“6j # 0},

= + = _ = ' .
Nls Ns 68, N1t NC 68 and N1:] N1 for the other j 8

Thus N1 stays between N and N' coordinatewise and differs from

N in two coordinates. Repeating this construction, the process

~

terminates in r s n steps since each successive Ni identifies

at least one more coordinate with N'. Define

~

m N m N

T1 = x Fjl"lj - x F ij, i = 1,...,r.
i=0 i=0 3
Since Ni differs from Ni-l in two coordinates, say 3 and t,

* *
the fact that (Tig) = (Ti) enables us to use Lemma 1 to obtain

N N,_ * N, N
(45) HIZH s “(FSi-ls X Fti 1t) _ (F is X F it)*H'

Applying Theorem 1 . to RHS (45), we note that, since each Ni

stays between Ei-l

in this application of RHS (42) are bounded below by As and At

and Ni coordinatewise, the denominators

+1

respectively. Since K is increasing, we further weaken this

application of the bound (42) by replacing dst by d. Thus

(46) [RHS (45)]2 s K(d)(Ni_1S - Nis)2(%- + 71—).
S C

39

Since Ni-lj = Nij except for j # s,t, RHS (46) is
m
-l 2
K d 2 A - N. .
()3.an mmj lj>
r
Since T = 2 Ti, by NS-SI and the above representation of RHS (46),
i=1
* 2 * 2 -1 2
(47) HT H s r )3 “'11“ s r K(d) zjjAj 21 (Ni-lj - Nil) .
Since E (Ni-lj - N11) = 6j and Since Ni-lj - Nij are of the

same sign for fixed j, the summation w.r.t. i in the last term
2
of (47) is bounded by oj. Since r s n the proof is complete.
Theorem 4. Let F0,F1,...,Fm pg internally connected by

chains with successive elements non-orthogonal and let
V .
d = v{dij‘Fi/‘(Fj}' Then

HT*H2 s % m K(d)(213|51‘)2 2i A;1.

‘ggggf. For any connected graph of finitely many vertices
there exists a vertex whose removal leaves the remaining graph
connected. We shall rename F0,...,Fm in such a way that
successive removal of F0,F1,... leaves the remaining connected.
For each i, let t(i) be 9 t(i) > i and Ft(i) ,KFi.

Given N and N' we consider the partition which differs

from either N (if 6 S 0) or N' (if 6 > 0) only on the

0 0

O-th and the t(0)-th coordinates, where the 0-th coordinate is

NO A N6 and the t(0)-th coordinate, compared to that of N or

u', is increased by ‘6 By weakening Theorem 3 on both K

o‘-

and the second denominator (l + the t(0)-th coordinate of the new

partition 2 At( ) + ‘601 2 A we see that the square norm of

* of the difference between the product measures associated with

0

40

the two partitions is bounded by

(48) 1((3’) 539% + X—1——).
0 t(O)

(i)
j

j-th coordinate at the i-th iteration of this process, we see that

Iterate the process. Letting 6 be the difference in the

6§1) = 0 for j < i and

(49) 6§i) = 6j +-Z {6:r)‘0 s r < i, t(r) = j}
for j 2 i. We also note that the 6st) above are disjoint sums
of 6's from {60’°"’5i-1}'

Since A(i), the minimum of the j-th coordinates for the two

:1

partitions at i-th iteration plus 1, is increasing w.r.t. i, the

bound correSponding to (48), further weakened by A(1) 2 A., is

i i

Since each iteration results in reducing one coordinate
difference to zero, the process terminates in m steps. By NS-

SI as in the proof of Theorem 3, we see that, by (50),

2
* v '
(51) HT “2 s m K(d) 2 6:1) (Al— + R—l—L
i i t(i)
. . 1 . . . (m)

The coefficient of X—- in the summation above is, With 6m = 0,

i

. 2 2
(52) 6:1) +-z {6:r) |t(r) = i}.

r

By (49) and the comment following it, complemented by 2 61 = 0,
we see that 6:1) and the 6ir) in (52) are disjoint sums of

{50"°"6i-1’6i+l"°"6m}’ Thus the maximum of (52) Over

41

all g is s (2%9L - 5:)2 +-(§%§L - 5;)2 s %i(z|o|)2, and the
proof is complete.

The hypothesis of Theorem 4 fails to hold if and only if
9 is dis connected.

Then either (i) there exists a component, i.e.

a connected set of factors, whose N-multiplicity differs from its

~

*
n'-multiplicity, in which case it follows easily that ”T H = l,
or (ii) every component has identical N- and N'-multiplicity,

in which case “T*H is simply related to the “(Tc)*H correspond-

ing to the separate components:
(53) v “(Tc)*H s HT*H s 2 “(TC)*H.
c c

where the second inequality follows from Lemma 1 and the triangular

inequality.

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