AUNIFORM CENTRAL LIMIT THEOREMFOR PARTIAL-SUM PROCESSES INDEXED BY SETS

BY

RONALD PYKE

DEPARTMENT OF STATISTICSUNIVERSITY OF WASHINGTON

SEATTLE} WASHINGTON

TECHNICAL REPORT #17JANUARY 1982

This work was supported in part byNational Science Foundation Grant MCS-78-09858.

Manus typed by

AUNIFORM CENTRAL LIMIT THEOREMFOR PARTIAL-SUM PROCESSES INDEXED BY SETS

BY

RONALD PYKE 1

UNIVERSITY OF WASHINGTON

IThis work was supported in part by the National Science Foundation,Grant MCS-78-09858.

ABSTRACT:

Let A be a family of subsets of the unit k-dimensional cube Ik and let

{Xi} be an array of independent random variables indexed by the lattice of

non-negative integral k-tuples. l·(jl •...• j k) . For each nz I and AEA

define the partial-sum Sn(A) to be the sum (normalized by n-k/2) of all

Xi for which 1In EA. By classical results the finite dimensional distri­

butions of the partial-sum process {Sn(A): AEA} are asymptotically normal.

It is proved in this paper that when appropriately smoothed. this process

converges weakly to a Brownian process under weak restrictions on A and

the moments of {Xi}' Applications of this result are made in later sections

to derive weak convergence results for Poisson and Uniform empirical processes.

AMS 1970 Suhjeat C14ssifiaation. Primary 60B10Secondary 60F05. 60J65

Key words and phrases: CBntral Limit theorem. inde:t:ed by sets. weak

oomierqenee, Poisson process. empirical process. partial-BU1I'I process.

1

1. Introduction.

The present paper considers the question of uniformity of convergence for

the Central Limit theorem in the context of random measures on the plane or

higher dimensional Euclidean spaces. Perhaps a point process type motivation

is most appropriate. Consider a large forest in which some insect infestation

is present. Suppose the forest is partitioned by a grid into relatively small

equal area sections and then measurements are taken within each section of the

amount of damage or number of insects or some other appropriate quantity. The

resulting matrix of observations is of the type considered here. For any

region A. not necessarily rectilinear. a measure of its damage could then be

given by summing over the sections that approximate A. If the measures of

disjoint sections are independent. the sum for large A would be approximately

Normal by the classical Central Limit theorem. The purpose of this paper is

to derive a Uniform Central Limit theorem for the normalized sums when viewed

as a process indexed by a large family of sets A.

Other situations in two or three dimensions are easily imagined with regard to

census data. mineral deposits. flaws in steel plates. and so on. In Section 4

below we apply the results of Sections 1-3 to obtain a Uniform Central Limit

theorem for Poisson processes. We then apply this result in Section 5 to

illustrate a method for deriving a Central Limit theorem for Uniform empirical

processes.

Following Pyke (1973). let {Xj: lEJk} be a set of independent and identically

distributed random variables where Jk denotes the set of k-tuples of positivekintegers ! .. (jl •.•••jk). Let I .. £Q,.,V denote the closed k-dimensional unit

cube. If one views Xl as a random mass at the point 1. then for any subset

2

Be Rk one may define the pa:rtial.-SUtrl signed measure

The purpose of this paper is to derive a Uniform Central limit theorem for

the partial sum processes {S{nA): A E A} where A is a suitable family of

closed subsets of lk.

Assume throughout that EX j • 0 and var{Xj ) . 1. For any Borel s~t B, let

B& denote the &-neighborh;od of A with ;espect to Euclidean distance.

Write B& for the "inner" neighborhood defined as the compliment of the

&-neighborhood of the compliment of B, namely B&. ({Bc)&)c. Write B{&)

for B&'B&, the &-annulus around the boundary aB of B. lebesgue measure

will be denoted by either ~ or \-1 as eonvenient.

The index families A considered in this paper are assumed to satisfy the

two conditions:

Al. There is a constant c>O such that for all &>0 and AEA,

A2. A is totally bounded with respect to the Hausdorff metric dHdefined by

0.1)

Recall that the set of all closed subsets of lk forms a complete and separable

metric space with respect to dH• (Cf. Debreu (1967) e.g.) let v{&) denote

the smallest cardinality of an &-net in A.

3

Write

( 1.2)

for the normalized partial sums. Then under the above assumptions.

Sn(A) has mean zero and variance n-kN(nA) where for any Be Rk N(B): '"

card{j: j €Bl.

When A is the class of lower crthants , [O.t] for t€ Ik"" "" ""

naturally write Sn(t) for S ([O.t]). Observe that then,....., n ~......,

cov(Sn(,t). Sn(~» = I[£.,t] n [£.~] I whenever n~. n,t€Jk.

one may more

In Kuelbs (1968)

it was shown for k"'2 and under certain additional moment assumptions. thatL

Sn - Z where Z'"{I(,t): ,t € I2} is a 2-dimensfonal Brownian Sheet; that is.

Z is a Gaussian process indexed by Ik with mean zero and

In Wi chura (1969). multidimensional versions of Kolmogorov and Skorokhod

inequalities are obtained which enable Wi chura to prove that Sn~ Z under

the finiteness of only the second moment. Here ~ denotes convergence in

law (equivalent to the weak convergence of the image laws) with respect to the

Skorokhod topology. The reader is referred to Pyke (1973. Sections 3.5 and 3.6)

for more detailed discussion of these and related results.

The purpose of this paper is to show that analogous limit theorems are possible

for more general index sets A. Problems of this nature were posed by the author

about six years ago and preliminary results were obtained in 1977 and 1978. These

results used the methods of Pyke (1977) and were unsatisfactory in that the

restrictions on the tails of the distributions. namely generalized Gaussian.

seemed to be unnecessarily strong. In the present paper the only conditions

are in terms of finite moments of orders that increase as the "size" of the

4

index families increase. It is still conceivable that a finite second moment

is all that is required. but I believe that the orders assumed here are nearly

optimal. In Erickson (1981). a detailed study of weak convergence in Lipschitz

spaces is made and applied to partial-sum processes. This stronger convergence

is obtained under generalized Gaussian assumptions for large index families and

under a p-th moment condition (p> 2) for smaller index families of similar

size to the orthants.

The limiting process in the general case. Z· {ZeAl: AEA} win necessarily

be the Brownian proaess inde:ed by A defined as a mean zero Gaussian process

with

cov(Z(A). Z(S» = IAnsl ; A.SEA.

Let C(A) be the set of all continuous real-valued functions defined on

(A. dH). It is known (cf , Dudley (1973» that if A is not "too large".

there exists a version of Z in C(A). For example. if

for some r< 1 and K>O. and all sufficiently small &>0. then

ZEC(A) is possible. In this context. the infimum of those r for which

(1.4) holds is called the exponent of metria entropy.

( 1.3)

( 1.4)

A particular class of index families A for which (1.4) is true was introduced

by Dudley (1974). To describe these. fix a> 0 and M> O. Then let

A =A(k.a.M) denote the set of all closed subsets of Ik whose boundariesa

are represented parametrically as continuous mappings of the k-1 dimensional

unit sphere. which have bounded (by M) derivatives of all orders up to but less

than a and for which the derivatives of highest order satisfy a Lipschitz

condition of order a - raj > O. (In this definition we

0.5)

5

use [a] for the greatest integer less than a.) Dudley (1974) shows

that for A. (1.4) holds for any r>(k-l)/a. For further discussiona

of A see Sun and Pyke (1982). A closely related and more simply defineda

family. R(k.a.M) say. of sets with smooth boundaries was introduced by

Revesz (1976). The exponent of metric entropy is also (k-l)/a for

R(k.a.M).

Other candidates for index families include Ck• the class of closed convex

subsets of Ik• and Pk•m• the class of all closed polygons inIk with

no more than m vertices. The exponent of metric entropy of the former is

(k-l)/2 (cf. Dudley (1974» and of the latter is zero which may be deduced

straightforwardly (cf. Erickson. 1981).

In order to derive the weak convergence of the partial-sum processes to Z

it will be necessary to place conditions on A like those above. to insure

that the limit ZE C(A) a.s. It will also be necessary. and at first more

surprising. to restrict oneselves to continuous versions of the partial-sum

processes themselves. It was not necessary to do this for the Wichura result

for which A was the relatively small class of lower orthants. The reason

that smoothing is needed in general is that with a larger A it may be

possible to have very close sets whose boundaries weave in and around the lattice

points j/n in such a way that the S -measures of the symmetric differences'" n

are not uniformly small. This was discussed in Pyke (1977) and a specific

example due to Dudley is included in Erickson (1981).

The natural continuous version of the partial-sum process is defined (cf. Pyke

(1973).4.3.5) as the random signed measure

Z (A) • n-k/2 X(nA)n

6

where. for any Borel set Be Rk•

X(B) • t j EJklBnCjl Xj'" '"

where C1 is the unit cube (1-1•.V. Thus X is a signed measure which

is absolutely continuous with respect to lebesgue measure x and for which

the density dX/dX is equal to Xj on Cj• Asimilar description could""also be given for Zn'

lemma 1.1 The finite dimensional distributions of the Zn-process converge

weakly to those of Z.

Proof. let both Sn and Zn be defined in tenns of the same array

{X.: j E J}. It will suffice to show that Z (A) - Sn(A) 1.> 0 for eachl "" n

AEA. since the finite dimensional distributions of Sn are straightfor-

wardly shown to converge to those of Z. To this end. write

Zn(A) -Sn(A) • n-k/2

t j : n-1Cjn ClA=G wj Xj ( 1.6)

where w1

= InAn C11 - IA(1/n). The number of summands in (1.6). say m,

can be bounded as follows. Each cube Cj is of diameter k1/2 and volume 1.

Since the union of those cubes n-1Cj which intersect CIA is therefore

1 1/2 ""contained in the annulus A(n- K ) . it follows that

where the last inequality uses Assumption AI. Thus since IWjl! 1•

•

7

as n-..... This shows that Zn(A) -Sn(A) 1.. 0 for each AU•• which in

turn implies that the asymptotic finite dimensional distributions of the

Zn-processes are the same as for the Sn-processes which are those of Z. 0

It is the fact that Zn(A) - Sn(A) need not converge to zero uniformly over

A which necessitates our emphasis upon the continuized version of the partial­

sum processes. The weak convergence of the Zn-processes pertains to (C(A).dH)and for this a tightness condition involving the modulus of uniform continuity

is all that is needed in addition to lemma 1.1. The study of tightness in

the next section will then complete the proof of

Theorem 1. let {X j : 1. EJk} be an array of independent. identically distri-'"buted random variables with zero mean and unit variance. If for O<r<1 and

-r ss » 2(l+r)/(l-r). 1n v(c.) =O(e ) as e..... O and E( IXll ) < -. then under

assumptions Al and A2 the normalized continuous partial sum process

{Zn(A): AEA} converges weakly on (C(A).dH) to {Z(A): AEA}. the

Brownian process indexed by A.

2. Tightness of the Zn-processes. As in Pyke (1977) let AF(e) denote a

finite e-net in A of minimal cardinality v(e). For any A.B or C in A

we will write A~. B~. or C~. respectively. to denote any member of AF(e)

within e of the given set. That is. dH(B. B~) s e and B~ EAF(e) for

example.

To establish Zn ~ Z it suffices in view of Lemma 1 to show that the

image laws of the processes are tight. To this end. denote the modulus

of continuity by

8

(2.1)

To establish tightness it is sufficient in view of Lemma 1 (cf. Billingsley

(1968). p. 55) to show that for every &>0 there exists 5>0 and an

integer no such that

We will obtain this result by detailed approximations that require Z to. n

be represented first as a telescoping series of values of Z evaluatednat elements in the approximating nets. This is described as follows.

The supremum in (2.1) is over an infinite set of values. Nevertheless.

each BE A can be approximated by a sequence of elements chosen from the

e-nets , To this end. choose 5j-0 and let 50· 5. For each Band CF F Fin (2.1) select Bo.• COjEA (OJ) for each j. When dH(B.C)<0/2 assume

F F JBo =Co Then

o 0

kn F F F Fw(Z ,0/2) :::ZJ'-l max{ lln(Bo )-Z (Bo ) 1+ IZn(Co.) - Zn(Coi ) I: B.C EA, dH(B,C)<0/2}n - j nj-1 ~ -1

+ supnZ (A) - Z (A~ )1: AEA}. (2.2)n n kn

Although card AF(5j) increases as OJ decreases, it is nevertheless finite

for every j and if the 0j_1 - OJ are sufficiently small we will be able to

get sufficiently small uniform bounds on each of the terms in (2.2). This

nesting approach was introduced by Strassen and Dudley (1969) and used subse­

quently by Dudley (1978) and Sun and Pyke (1982) in applications to empirical

processes. For this approach to work here we will need to be able to choose

{OJ} and a sequence of bounds. {~j}' so that for some kn

9

and

unifonnly for n~no for a preassigned &>0. Similar results will follow

for the reversed differences B~ ..... B~ and for the analogous differencesJ-l j

for the C-sets. In going from (2.2) to (2.3) we have used the fact that

Zn is finitely additive so that for any two sets.

The summation in (2.3) can be replaced by one slightly simpler in fonn.

namely.

{2.3}

(2.4)

(2.5)

{2.6}

since v(Oj) ~ v(Oj_l)' where Bj represents any set which contains no

sphere of diameter OJ_I' Note that by construction. dH{B~f B~j_lh 2o j _l •

Before approximating the summands in (2.6) we first truncate the array

{Xl}' For given n. truncate each Xj at ~n and write Xi{~n) for the

truncated version of Xi' Here. trunc;tion means that the truncated values

are replaced by aero. Choose ~n so that

in order to insure that

P[X. s X.(~}: for all j ~ n 1] ~ 1 - &.l l n '" '"

(2.7)

10

Set t n = n-k/2'tn• For each B. introduce the coefficients bj=n-k/2InBnCjl

...,so that we may write

In(B) = 1:i. E Jk

b.i X.i'

Let l~r denote the partial-sum process determined by the truncated array;

thus

To minimize complications assume for now that the truncated variables have

mean zero. At the end of the proof we will check that this assumption can

be dropped.

Recall (cf. Bennett (1962» that Bernstein's Inequality states that if S

is a sum of independent mean zero r.v.'s each bounded by M. then for all

P[S:>tl < exp{-t212(a2+Mt/3)}

where 0 2 = var(S). For the sum l~r(B) in (2.8) we have

and

Thus for the Bj of (2.6). IBjl ~ 2coj_1:=C'Oj_l by AI. Bernstein's

Inequality may then be applied to show

p[l~r(Bj) :> ~jl < exP{~j/2(coj_l + t n ~j/3}

= eXP{~j/25j_l(c'+ t n ~/35j_l)}'

(2.8)

(2.9)

11

We choose geometric sequences

Substitution of (2.9) and (1.4) into (2.6) shows that (2.6) is bounded by

kLj~l eXP{2Kojr - "jt2oj _1(c'+tn ,,/30j _1) }

(2.10)

Notice first of all that in order for this series to converge as kn . - ,

we must have 1-2b~r or b~(l-r)/2. Such a choice is possible since in

order that Z E C(A) a.s , we have assumed r< 1. Secondly, if we write the

exponent in (2.11) as

it is seen that in order to insure convergence the term in parenthesis must

remain positive. This will be satisfied in particular if we allow

(2.12)

t ~(b-l)kn ~ - (2.13)n

but postulate that

t ~-(r+b)kn ~ Y (2.14)n

for some suitably small constant Y> 0 to be specified later. Notice that

if 1-2b>r, (2.14) implies (2.13).

sIn Theorem 1 it is assumed that E( xII ) < - for some s> 2(l+r)/(l-r).

This implies that

Prlx11 > xl • o(x-s)

."'"

The truncation condition (2.7) is then satisfied if we choose {tnl so that

(2.15)

12

or equivalently

t • n-( l/Z - l/s}k An n

where {An} is any positive sequence diverging to +... Together with

(2.14) this implies that

exp{-(1/2 -l/s)k In n + In An + (-1n e)(r+b)kn}

... y.

It therefore suffices to choose kn to satisfy

kn • (-In B)-l(r+b)-l {(I/2 - l/s)k In n -In An + Yn}

for some choice of yn ... In Y that makes kn a positive integer.

Another restriction upon kn is implicit in (2.4). For the truncated process

SUP{lztr(A'A~k )1: A€A} s n-k/z-r sup{\n(A'A 5k }I:A€A}n n n n

~ t n nk c 5kn

(2.I6)

(2.17)

by AI. To satisfy (2.4) we will require that

kt nk ~ n ... O. (2.1S)n

Substitution of (2.16) and (2.17) into (2.1S) shows that An and Yn

must allow the divergence to - .. of

kIn(tnnk~ n} :: In \ + (1/2 + l/s)k In n + (1n ~}kn

·In Xn

+ (I/2+l/s)k 1nn- (r+b)-1{(I/2-1/s}k1nn-lnXn+Yn

• -(k In n}{(1-r-b}/2 - (1+r+b}/S}{r+b}-l + {l+(r+b}-ll1nA + (r+b}-ly .n n

13

For this to happen it suffices to choose lnA • o(ln n) and to choosen

s so that the coefficient of -k ln n is positive; that is,

s> 2(l+r+b)/(l-r-b).

Since this bound is an increasing function of b we can always find

b >0 to satisfy (2.19) if

s > 2(l+r)/(l-r).

This latter inequality is the assumption made in Theorem 1.

It remains now to show that for the above choices of ~,b, s, k ,\ andn n

Yn, it is possible to choose values for the remaining parameters 0, ~

and Y so that (2.3), (2.4) and (2.5) hold for any preassigned E >O. Since

a choice of ~ sufficiently small will insure that (2.5) holds.

Expression (2.4) holds by construction since (2.18) insures that the event

never occurs for large n for the truncated processes, while (2.16) in

(2.7) insures that the probability of equal truncated and non-truncated pro­

cesses can be made arbitrarily small. To check (2.3), observe that the

exponent of the bound (2.11) when written in the form (2.12) is

-r -rj { or ~(r+b)j _ I}-2Ko ~ 4K~+0(1)' t n

where 0(1) indicates here that as j increases, the term decreases to

its value for j=k, which value then converges to zero as 0---- by (2.13).n

Also, by (2.14) the term ~(r+b)j/tn decreases to its value for j=kn

(2.19)

(2.20)

(2.21)

14

which in turn converges to l/y. Therefore. if we choose Y-5r/8K~.

the parenthetical term in (2.21) is positive and bounded away from zero

uniformly in 5. Consequently the sum in (2.3) remains bounded as

n... -. The bound on these SllllS can then be made arbi trarily sma11 by

choosing 6 sufficiently small. since the only effect of 6 after our

choice of y is as the front multiplier in (2.21).

In the above derivations we assumed that lJn:· EX,t('tn) =O. If this is

not the case use the bound

Since for any B

the last t~rm in (2.22) is bounded above by 2lJ nnk/2c5 because of AI.

But in view of (2.15)

Thus by (2.16)

since s > 2 and A ... -. This shows that the set of deterministicn

functions {E(ztr(.): n~11 is tight. demonstrating that our previousn

assumption was made without loss of generality.

3. Extensions to non-identically distributed arrays. In the preceding

derivation of tightness for independent and identically distributed summands.

the assumption of a common.distribution was used only in the computation of

(2.22)

15

variances such as in (2.8) and in the derivation of the truncation level

T as in (2.16). In what follows we describe briefly a natural general­nization of these two steps that applies to non-identically distributed

arrays. To this end let {X J': j EJk n nlk} be an array of independentn '"

random variables with EX . ="'0 and var(X .) .. c? <. for each j.nJ nJ nJ '"

Define for each n ~ 1 and'" BEBk '" '"

An (B) = 2:j In-1Cj n BIO'~j :'" '" '"

Then An is a easure. Let us assume that there exists a measure A

for which

lim A (B) = A(B). uniformly for BE Bkn"'''' n

and .o.(B) ~ ColBI for all BE Bk and some constant Co' (These conditions

could have been stated in terms of dA/dX. t ,e , the function O'~(t) =O'~i

if nt EC.. The given statements are in a more natural form for applicationJ

here.) I~ place of (2.9) one can now write

var(Z~r(B)) ~ var(Zn(B)) = n-kZjlCjnnBl2 O'~j'" '" '"

for n sufficiently large. which is the inequality needed at that point

in the proof.

For truncation. assume in place of (2.15) that

Ln(x): = n-kzj PI!Xnjl>x]" O(x-s).

'"

The proof of tightness then continues without further changes.

16

The analogue of lemma 1 for the assumptions of this section is the following,

derived straightforwardly as before by use of classical central limit theorems.

lemma 2. The finite dimensional distributions of Zn converge weakly to

those of wA - {wA(A): AEA}. a mean zero Gaussian process wi th

cov(wA(A). wA(B»- Il(AnB). A.BEA.

Notice that as a consequence of the convergence Z J6> wA. one obtainsn

the fact that wA EC(A).

4. An Application to Poisson Processes. let N denote a homogeneous

Poisson (1) process defined on Sk. so that N(B) is a Poisson random

variable with EN(B)"' IBI. For any x> O. write

that N). is a homogeneous Poisson (A) process. Consider the normalized

process {W (A): AE A} defined byn

Wn(A) "' n-1/ 2[Nn(A) - nIAI]. (4.1)

Clearly the finite dimensional distributions of Wn converge weakly to

those of the Brownian process Z. This holds even if A"' Sk. To show

that W J:.. Z (where J:.. I1lJst be clarified since W ~ C(A» we willn n

produce a matrix array {Xnj} for which the corresponding In-process is'"close to Wn• To this end set

Xnj

"' n(k-l)!2{N(nl/k-1C1) - n1-k}.1 EJk.

which is a Poisson (n1-k) random variable. normalized to have mean 0

and variance 1. Since these arrays {Xnj} satisfy the conditions of'"Sections 2 and 3. the following result obtains.

(4.2)

17

Theorem 4.1 If {Zn(A): A€Al is the (continuous) pari tal-sum process

formed from the array (4.2). then for any A satisfying AI. A2. and (I.4)

for any 0 < r < 1. Zn"!:' Z on (C(A). dH).

We consider now the difference between Wand Z. Although we stressedn n

at the outset that one cannot in general obtain limit theorems without

smoothing the partial-sum processes. it is possible in this Poisson case

to show that Wn and Zn are uniformly close because the expected number

of discontinuities (atoms) of the Poisson process in Ik is only n. much

less than the number nk of summands. To verify this. write as in (1.6)

= n-1/2rj{ICj n nAINn(n-1Cj) - Nn(n-1Cj n A)}1"V I""v ........ ~

Thus, if An denotes the 'rectangular' approximation to A defined by

A = U{n-1C.: n-1C.nA -; ~},n J J

'" '"

Iz (A) - W(A>I s n-1/ 2N (A 'A).n n n n

Now A"'-AcA5..... A for any 5>k1/ 2n-l• the diameter of each cuben

-1 1/2 -1n Cj • Thus for any 5> k n

P[suPAEAIZn(A) -Wn(A)1 >c] s P[SUPA€A Nn(A5, A} >cn

l/ 2J

s P[maxA€l(cfn(A25'A)

1/2s v(5) P[Y2c5n >c n l

1/2J> c n

(4.2)

where vee) is the cardinality of AF (5) and Y2c5n denotes a Poisson-

18

2c5n random variable. If we assume v(S) is large and staisfies (1.4)

for some rE (0.1) then a small exponential bound for the upper tail of

the Poisson distribtion is needed in order to make (4.2) small. To this

end we prove

Lemma 4.1 If YA is a Poisson-A random variable. then

P[YA> bAa] :::exp{b(l-lnb)Aa - b(a-1)Aa In A- A}

for any b > 0 and a> 1 satisfying bAa> A.

Proof. The proof is standard. By Chebichev's Inequality. for any t>O.

tY a a (t )P[YA>bAa) = Pte A>e t bA]::: e-t bA +A e -1.

This bound is minimized when t = In(bAa- 1) . so that whenever bAa-I> 1.

which is the stated result.

To apply this to (4.2). set A=2con. Write s=n- 1+q so that A=2cnq•

Take b = e:(2c)-a so that Aa = (2c)anqa must equal (2c)anl12. Thus

a = 1/2q and the bound of Lemma 4.1 becomes

-1/2q 1/2 1/2 qexp{t(l-ln t(2c) )n - c(1/2q-1)n In n - 2cn }.

o

Substitution of this into (4.2) yields a bound of

r( i-q) 1/2 -1/2q 112 qexp{Kn -c(1/2q-1)n lnn+t(l-lnd2c) )n -2cn}.

Clearly this bound can be made arbitrarily small for n sufficiently large

if r(l-q)::: 1/2 and 2q< 1. (Notice that the condition bAa> A in this

application becomes tn1/

2> 2cnq• which is satisfied for all sufficiently

large n if q<1/2.) Thus by choosing q tosatisfy 1-1/2r<q<1/2.

19

which is always possible since r< 1. one can make (4.2) arbitrarily small

for all sufficiently large n. This establishes the fact that

(4.3)

for all £ > O. This may be written as PA(Z. Wn) 1. 0 where

PA(f.g) '" sUPA EAlf(A) - g(A) I is the uniform metric on A.

Since all moments of a Poisson random variable are finite. it follows from

Theorem 1.1 that for all rE(O.I). Zn"!:' Z. Since ZnEC(A). ne L,

and ZEe(A). a complete and separable metric space. the convergence in

law • ..!:. • of our previous theorems is defined by saying that Eg(Zn) +Eg(Z)

for all continuous bounded functions g: C(A)+R'. For processes such as

Wn that do not take values in a separable metric space. we use ..!:. to

mean that Eg(Zn)+Eg(Z) for all continuous bounded real valued functions 9

for which g(Zn)' n~ 1. and g(Z) are random variables. [cf , Pyke and

Shorack. 1968). For Wn we use the supremum metric PA on any approved

sample space that contains C(A). Theorem 4.1 and (4.3) then verify the

following result.

Theorem 4.2 If Wn is defined as the normalized Poisson process of

(4.1). then for any A satisfying AI. A2. and (1.4). Wn ..!:. Z with

respect to PA'

5. An Application to Uniform Empirical Processes. Let Un'" {Un(A): AEA}

be the Uniform empirical process indexed by A. That is. if Vi•.•.•Vn

are independent Uni form (I k) random vari ab1es and Fn '" n-1 (6 V+' •• +6V )1 n

is the empirical meausre where 0x(A) '" 1 or 0 according as x EA or

xl-A. then Un(A) '" ni/

2(F

n(A) - IAI). If Nn denotes a Poisson -n process

20

as in the previous section, define

Since {Nn(U k): t?O} is a non-homogeneous Poisson process with

E[Nn(tI k)] • ntk• it is clear that T~/k is a r(n+1: n) random

variable so that T~/k 1. 1 as n·.... Also. therefore. Tn 1. l.

(Actually, by the strong law of Hsu and Robbins. we could claim

Tn~ 1 since the fourth moments of Exponential random variables

are finite.)

Consider the process U~ defined on A by

This process has the same finite dimensional distributions as does Un'

a fact we leave to the reader. To show weak convergence, we use Theorem

4.1 and (4.3) in conjunction with Skorokhod's construction (namely, if

Zn J:,. 2 on a complete separable metric space, (M.d) say, there exists

equivalent processes. Z~. 2* say. on a common probability space for

which d(Z~. 2*) .!.:.!:..,. 0,) to show that without loss of generality we

may aSSume that PA(Wn, 2) 1.0. Since Tn 1. 1 it follows that for

the strongly convergent versions

uniformly in A€ A where U is the 'tied-down' Brownian process associated

with 2. Note that the mapping A+tA is continuous with respect to dW

This is enough to establish

Theorem 5. If Un is the Uniform empirical process and A satisfies

AI. A2 and (1.4). then Un J.U with respect PA'

(5.1)

21

For more direct approaches to this result see Dudley (1978) and Sun

and Pyke (1982).

6. Concluding Remarks. I have focused in this paper on large index

families satisfying (1.4). For Many smaller classes such as the lower

orthants or Pk,m (see Section 1) the analogue of (1.4) would be

for constants K>O and r>O. For example, for the orthants, rak

and for Pk,m' rakm. In the case of (6.1) it should be possible to

modify the arguments of Sections 1 and 2 to establish the uniform Central

Limit theorem under the fi niteness of second lIIOIIlents only. This prot>1em

will be considered separately.

The full Central Limit problem may also be considered, in which non-Gaussian

limiting processes would be involved. Since these processes are not in

C(A), considerable care must be exercised concerning the discontinuities

near the boundaries of the sets in A, even if one still works with 2n,the smoothed version of the partial-sum processes. The methods of this

paper do not seem to be applicable to the general infinitely divisible

situation. They may however be applicable in the possibly more applicable

direction of dependent arrays which satisfy appropriate mixing conditions.

* * * * * *

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