ON STRONG LAWS OF LARGE NUMBERS FOR ARRAYS OF ROWWISE INDEPENDENT RANDOM ELEMENTS

VOL. 16 NO. 3 (1993) 587-592

ON STRONG LAWS OF LARGE NUMBERS FOR ARRAYS OF ROWWISE INDEPENDENT RANDOM ELEMENTS

ABOLGHASSEM BOZORGNIA

Department

ofStatistics Mashhad University

Mash_had,

Iran

RONALD FRANK PATTERSON

Department

of Mathematics and

Computer

Science Georgia

State

University

Atlanta,

Ga.

30303

U.S.A.

ROBERTLEETAYLOR

Department

ofStatistics University ofGeorgia Athens,

Ga

30602

U.S.A.

(Received March 19, 1992 and in revised form April 7, 1992)

ABSTRACT. Let {X, }

^beanarray ofrowwiseindependentrandom elementsinasep- arable Banachspaceof typer, 1

_<

r

_<

2. Complete convergenceofn^]/p

X.

^{to 0}

k=l

0

<

p

<

^r

<

2 is obtained when sup

EIIX.II ^{o(), >}

^{0 with}

1<<.

v

) >

^o

+

^1.

^An

application todensityestimation isalsogiven.

KEY WORDS AND PHRASES.

ltadom

clement, ,trong

^l,

o.f lr#e number, comzlet

convergence,

Rademacher

t/pe rpce.

1980

MATHEMATICAL

CI,AqqlFICATIOV

CODE.

60BI

1. INTRODUCTION

AND PRELIMINARIES.

Let (, I1.11)

^beareal

separable

Banachspace.

Let (ft, 4, P)

^denoteaprobability space.

A

randomelement

X

in isafunction from

t

into which is

4

measurablewithrespect of the Borel subsets

B().

The

p

absolute moment ofarandom element

X

is

EIIXll

^where

E

isthe expectedvalue of the random variable

IIXII

^{p. The}expected valueofarandom element

X

is defined tobe the Bochner integral

(when EIIXII ^{< o)}

^{d i denoted by}

EX.

The concepts of independence and identical distributions for real-valuedrandom variablesextenddirectly to

’. A

separable Banachspace^issid to beof

(Rademacher)

type r, ¹

_<

r

_< _2,

ifthereexist aconstant

C

suchthat

PATTERSON R.L. TAYLOR

E X

<C

EllXII

k---I k----1

for all independent random elementsX1,...,

X,

with zero meansand finiter^’h moments.

Every

separable Hilbert space and finite dimensional Banachspace isof type 2.

Every

separableBanach spaceisatleasttype 1 while and

L

spacesareoftype

rain(2, r)

^for

r>l.

Throughout

this paper

{X,

¹

<_

k

<_

n,n

>_ 1}

^{will denote rowwise}independent randomelementsin such that

for allnand k.

(1.1)

Themajor resultsofthis paper show that

n

1/ E ^X,t

^{---,0 completely}

^(1.2)

where completeconvergenceisdefined

(as

ⁱⁿ

Hsu

andRobbins

[1])

by

n=l k----1

> ] ^<

^o

^(1.3)

for eache

>

^0.

ErdSs

[2]

showed that for an array of^i.i.d, random variables

{X,k}, (1.3)

holds if andonly if

E[XI] <

^{o. Jain}

[3]

^{obtainedauniform} strong lawoflarge numbersfor sequencesofi.i.d, random elements in separable Banach spaces oftype2 whichwould yield

(1.2)

with p 1 for anarrayofi.i.d, random elements

{Xn}

^{in a}type 2 space.

Woyczynski

[4]

showed that

n/- . ^X,

^{.--,0 completely}

^(1.4)

for any sequence

{X,}

of independent random elements in a

Banach

^{space of type} r, 1

_<

p

<

r

_<

2 with

EX

^{0for allnwhichisuniformlyboundedby a}random variable

X

satisfying

EIX[P <

^oo. Recsll thatan array

{X,k}

^ofrandom elementsissaid to be uniformlyboundedby arandom variable

X

iffor all n andk and foreveryreal number

P [[[X,,[[ > ,] _< P[IX[ > ]. (1.5) Note

that^i.i.d, random elementsareuniformly boundedby

[[X, [[.

Moricz,

Hu,

and Taylor

[5]

showed that Erd6s’ result could be obtainedby replacing the ^i.i.d, conditionby the uniformly bounded condition

(1.5).

^{Taylor and}

^Hu [6]

^obtainedcomplete convergencein

type

rspaces, 1

<

r

_<

2for uniformly

bounded;

rowwise independent random elements.

Theresults ofthis paperrelaxes theassumptionof uniformly bounded random elements in Taylorand

Hu [6]. ^Moreover,

^amajorapplicationof themainresult ofthis paperis

indicated for kernel densityestimatorswhere uniformly bounded randomvariablescannot be asumed.

2. MAJOR RESULTS.

The following lemmafrom Woyczynski

[4]

^{will be used in} obttdningthe major

result,

Theorem 2.

LEMMA

1.

Let

1

<

r

<

2 andq

>

1. The followingproperties areequivalent:

(i)

isoftyper

(ii)

^{There exists a} Csuch thatfor

MI

independentrandom elements X1,...

,X,

ⁱⁿ

with

EXk O,

mad

EIIXII ^<

^oo,

^&

^1,2,...,n

Ell x I1’ ^{< CE} IIX

k=l k=l

THEOREM

2.

Let {X,k }

^{be an}array ofrowwisehadependentrandomelements haaseparableBanach space oftyper. If

EXn

^{0 and}

sup

EIIX.II ^{O(n), >_ o} (2.1)

l<k<n

nl/,

X,,t

0 completely.

k=l

PROOF:

Lete

>

^0begiven.

By

Markov’s inequality

n=l k=l n---1

v

k=l

n,l^p

Xn,

By Lemma

1 andHSlder’s inequality,

(2.2)

--x E IIX.ll

< C1

n----1 k----1 k---I

1,iv/r_

< ca

_n’/,

EIIXII

n----1 k----1

< ca

^{n/ .n up}

^llX.ll

n=l l<_k<_n

----’--I

1

n----1

A. BOZORGNIA, R.F. PATTERSON AND R.L. TAYLOR since

>

1

+ .

^Therefore,

1

X

^--.0 completely.

=I

REMARK

1.

For

vluesof pand r,1

_<

p

<

r

_<

2,it follows thatv

>

^2.

Moreover,

^asp andr move dose to eachothervincreases without bound.

However,

for certainvalues of p strictly less than one, avalue ofv 1 ispossible toobtain completeconvergence.

To

see this letp=

,

^{r landa 0.}

^It

^followsthat

_v(-) ^v(3-1)=

^2v

^>

^{l implies}

thatv

> 1/2. ^However,

^{the proofof}Theorem 2 requires that v

>_

1. Thus, v 1 is the smallestmoment necessary (givensuitable conditions on p,r and

a)

^{toobtaincomplete} convergence,viaTheorem2

REMARK

2.

Thecondition sup

EI[X,kI[" O(n’)

^issomewhatstronger than

(1.5)

used by Taylor

_<k_<-

and

Hu [6]. However,

^thebound in eachrowincreases as n oo whichisa substantia/

improvement inTheorem4 of Taylor,/Ioricz and

Hu [5].

This substantil improvement willbeillustrated inExv.mple I.

An

immediatecorol/arytoTheorem 2isobtainedfori.i.d, random elements.

COROLLARY

3.

Let {X,)

^{bean arrayofi.i.d, random elementsin aBanach}

space oftypersuchthat

EX11

^0.

^Let EIIXll ^<

^{o where}

( ) ^>

^1,0

^<

^p

^<

r

_<

2. Then,

1

X --

⁰ completely.

1/-

k---1

REMARK

^3.

The moment condition in Corollary 3 can be considerably smaller than the moment conditioninTheorem6of Taylor and

Hu [6], (see

^Remark

1)

^{butingeneralwillbemuch}

larger.

3.

EXAMPLE

1.

Let X1,..., X,

be^i.i.d, randomvariables withcommondensityfunctionf. Thekernel estimatorforfwithconstant bandwidths

hn

^isgiven

^by

1

K(’-X’) ^(3.1)

y() __

where

K

is abounded(integrab]e)kernel with compvt support

[a, b]

andthe sequence

{h}

^{is boundedand} monotonicaydecreasing to 0 ^{as n}

-

^{o0. Let}

^X=

^{be defined as}

follows:

Since the sequence

{X,}

^isi.i.d., it follows that

{X,,k

^k 1,2,... isi.i.d, for eachn.

Verification of Condition

(2.1)

^dependsonthe choice of

K,

thebandwidthsequence

{h,}

andtheparticularBanachspace. Typically,h,

O(n -’)

^where0

<

^d

< 1/2. ^To

^illustrate

the applicability ofCondition

(2.1),

^considertheBanachspace

L’,

1

<

^r

_<

2. Thenfor eachk andn

(

_< C’ h.’(

-")/"

_< C’n’"("-l/’.

Sinced

< 1/2

^andr

^>

^{1,v canbe chosensothat}

,up

llX.ll ^{o( )}

and

v

-

^>c+1 by lettingp=landc=dv ^r-1

r

Verification of

(2.1)

^follows easily for

Lq,

q

_>

2, since they are of type 2. Thus, n^-1

⁼ Xnk

--* 0 completely or

(nhn) - ^{= K t-X h. -E K t_-X h.}

^{--, 0}

completely.

Hence,

consistenc,

for(3.1, followssince(ha,-(K(f. -X)) h.

^--,

f(t)

by trtutitionaltechniques.

ACKNOWLEDGEMENTS.

This research was supported in part by the National Foundationunder contract

No. DMS

8914503 while the second authorwas at the

De- partment

of

Statistics,

University of Georgia,

Athens, Ga.

The researchfor the first and thirdauthors wasmainly completedwhile at the

Department

of Statistics, University of Georgia,

Athens, Ga.

REFERENCES

1.

Hsu, P. L.,

and Robbins,

H. (1947).

^Complete

Convergence affd

the

Law

of

Large Numbers,

Proc.Nat.Sci.

U.S.A.,

33,p. 25-31.

2.

ErdSe, P. (1949). On

aTheorem of

Hsu

andRobbins,

Ann.

Math. Statistics, 20, p. 286-291.

3. Jain,

N. C. (1975).

Tail Probabilities for

Sums

of

Independent

Banach

Space

Random Variables,

Z. Wahr.V.Geb, 33,

p. 155-166.

4. Woyczynski,

W. A. (1980). On

Marcinkiewicz-Zygmund

Laws

of

Large

Numbers,

rob.

^andMath.Statist_.,1, p. 117-131.

5.

Hu, T. C.,

Moricz,

F.,

and Taylor,

R. L. (1986). Strong Laws

of

Large

Numbers for

Arrays

ofRowwiseIndependent Random Variables,

Acta

Math.

Hung.,

54,

p. 153-162.

6. Taylor,

R. L.

and

Hu, T. C. (1987). Strong Laws

of

Large

Numbers for

Arrays

ofRowwise Independent Random Variables,

Intemat.J..Math.and Math

Sci.,10, p. 805-814.