ON THE WEAK LAW OF LARGE NUMBERS FOR NORMED WEIGHTED SUMS OF I.I.D. RANDOM VARIABLES

Internat. J. Math. & Math. Sci.

VOL. 14 NO. (1991) 191-202

191

ON THE WEAK LAW OF LARGE NUMBERS FOR NORMED WEIGHTED SUMS OF I.I.D. RANDOM VARIABLES

ANORI ^AOLER

Departmentof Mathematics IllinoisInstituteofTechnology Chicago, Illinois 60616U.S.A.

ANDREW ROSALSKY

Department^ofStatistics UniversityofFlorida Gainesville,Florida 32611U.S.A.

(Received March 14, 1990)

ABSTRACT. For weighted sums n

a:Y:

of independent

ancJ

identically.distributed random ^variables

,IJ

(n l/

=1 p

{Yn, n>l},

^a general weak law of

rrge

numbers of the form ^aY:-vn bn^{---,0 ts} estabhshed where

\=

1J

^/

{Vn, n_> 1}

and

{bn, n>_ 1}

^arestatableconstants. Thehypothesesrevolve botli thebehaviorofthe tad of the distribution of

IYll

^{and the}

grOWnbehavlors

^ofthe constants

{an, n_> 1}

^and

{b

n,

n_> 1}.

Moreover,aweak law isproved for weightedsums

a;Y;

^{indexedbyrandomvariables}

{Tn, n>_ 1}. An

exampleispresented

"1

whereinthe weak law holds but

tl’trong

law fails therebygeneralizing^aclassical example.

KEY WORDS

AND

PHRASES.

Weightedsumsofindependent and identically distributed random variables, weak lawoflargenumbers,convergence in probability,random indices, strong law oflarge numbers,almost certain convergence.

1980AMS SUBJECT

CLASSIFICATION

CODES. Primary60F05;Secondary 60F15.

I. INTRODUCTION.

Let

(Y, Yn, n_ I}

be independent and identically distributed

(i.i.d.)

random variables defined ona probability space (f/,s$,p), ^{and let}

(an, n_1}, (Vn, n_1},

^and

{bn, n_l}

beconstantswith an0, bn>0, n_l. Then

(anYn, n_l}

is said toobey the

general

weak lawoflargenumbers

(WLLN)

^{with centering}

constants

(vn, n_l}

^and norming ^constants

(bn, n_1}

^{if the} normed and centered weightedsum

a.Y.

n)/bn

^hasthe weak limiting behavior j=l

n

j=t

0 (1.1)

bn

where denotes convergence in probability. Herein, the main result,Theorem 1, furnishes conditionson

{an, n_>l}, {bn, n_>l},

and the distribution of

Y

whichensurethat

{anYn, n>_ 1}

^{obeys theWLLN}

(1.1)

for suitable

{vn, n_>l}.

It is not assumed that Y is integrable. Of course, the well-known degenerate convergencecriterion

(see,

e.g., Loire

[1,

p.

329])

solves,intheory, theWLLN problem. Theadvantage of employingTheorem lies inthe factthat,in practice, its conditions

(2.1), (2.2),

^and

(2.3)

^aresimpler and more easily verifiable than the hypotheses ofthedegenerate convergencecriterion. Jamison et al.

[2]

had investigated the

WLLN

problem in thespecialcase wherean>0,

bn

n

a;, ^n>

^{1, and max a.}

^O(bn).

j=l _<j_<n

A. ADLER AND A. ROSALSKY

Conditionsfor

{anYn, n>_ 1}

^toobey thegeneral stronglaw of large numbers

(SLLN)

n j--1

bn

0 almost certainly

(a.c.)

had been obtained by Adler and Rosalsky

[3,4]. In

Section5,anexample illustrating Theorem ispresented andthe correspondingSLLNisshown to fail.

TheWLLN problem isstudied in Theorem 2 in themoregeneral context of randomindices. More specifically, let

{Tn, n_> 1}

be positive integer-valued randomvariablesand let 1_<an--,o ^{beconstantssuch} that

P{Tn/an>A} o(1)

^forsomeA>0. Theorem 2provides conditions for

Tn

j=l

V[Cn]

_0,

b[an]

where the symbol

Ix]

denotes the greatest integerin x.

As

will become apparent, Theorem 2 ofKlass and Teicher

[5]

and Theorem 5.2.6 of Chow and Teicher

[6,

p.

131]

^provided, respectively, the motivation for Theorems and 2 herein. Moreover, our Theorems and 2areproved usinganapproachsimilar tothat of theearliercounterparts.

Some remarks about notation are in order. Throughout, ^a sequence

{Cn, n>_l]

^{is defined by}

cn

bn/lanl, n_>

1, and the symbol C denotesageneric constant

(0<C<oo)

^{which is} not necessarily the same onein eachappearance. Thesymbols

Unl"

^or

Unl

^areusedtoindicate that the givennumericalsequence

{Un, n>_ 1}

ismonotone increasingormonotonedecreasing,respectively.

2.

A PRELIMINARY LEMMA.

The key iemma inestablishingTheorems and 2 willnow be stated and proved. Itshould be noted that the conditions

(2.2)

^and

(2.3)

^areautomatically satisfied for the standard assignment ofan-l,

bn--n

n>l.

LEMMA. If

nP{lYl>cn} o(I) (2.1)

andeither

then

al

2EY2I([YI _<Cn) o(bn 2).

j=l

PROOF. Noteatthe outset thatc

n’["

undereither

(2.2)

^or

(2.3)

^andthat

(2.3)

^ensures 2 ^19.\

a.

obfi). ^(2.4)

j=l Thus,

(2.4)

holds under either

(2.2)

^or

(2.3).

{Bnk,

0_<k_<n,

n_> 1}

^by

Let c

0=0

^{and d}

^n=cn/n

^{n>_l. Define an array}

WEAK hAW OF LARGE NUMBERS FOR NORMED WEIGHTED SUMS 193

2 2

nnj=l J\

^k

0 for k=0, n,n>l.

forl<k<n-1,n>2

and

Itwillnowbeshown that

{Bnk

0_<k_<n,

n_>l}

isaToeplitzarray,thatis, n

k:__oIBnkl ⁰⁽⁾

Bnk0

^asn--,ofor all fixedk>0.

Clearly

(2.4)

^entails

(2.6).

Toverify

(2.5),

^{note that}

Bnk>0,

0<k<n,n>l,since

Cn’.

^Nowunder

(2.2),

^for all n>2,

n

2’/n-1 ((k+l)2-k2)d(

< (12 bn ^j__laj )lkk=

^k

)

^(sincedn|)

2/n_1

₂

^\

<- 3n ^{a ldk =O(1)}

andso

(2.5)

^{holds. On}the otherhand,under

(2.3),

n 2

dn"

^and

a. ^< Cna2n

^n>l.

j=l Then for alln

>

1,

Thus for alln

>_

2,

n

j=l Cn

C_C__

n

^{Bnk <} (_ ^(nl (k+l)2d2k+l-k2d)

k=0

\ndnJ\k=l

^k

\nd/\k=l"

C2 ((k+l)dk+l ⁺

^C

),k__ldk+, )

Cnd2n ^2C(n-1)d2n (since tint)

-< +

nd2n

0(1)

andagain

(2.5)

holds thereby proving that

{Bnk,

0<k_<n,

n>_ 1}

^{isaToeplitzarray.}

Thenby

(2.1)

^{and theToeplitzlemma}

(see,

e.g.,Knopp

[7,

^p.

74]

or

Love [1,

p.

250]),

Next,

notethat

n

k=0BnkkP{IYl>Ck ^{o(1). (.7)}

I

ai2Ey21([Y[<cn)

a2 ^Ey21(Ck <lVl<Ck)

2n2

^{j=l k=l}

j--’’l k’--1

^{x x-}

IYI <ck

_i n n

n n-I ₂

1_..

_.laj2(cp{.y,>0}. c2nP{,y.>cn} ₊ k__l(Ck+l_C)p{,y.>ck})

b2n

^j_-

(bytheAbel summationby

parts" hmma)

n 2

n1/c.4-1-

^k

<

bn

j=l

k’--l

^k

E

n

BnkkP{lY[>ck} ^{+ o(1)}

o(I) (by (2.7))

thereby proving the Lemma. ^[]

3. THE MAIN RESULT.

With the preliminaries accountedfor, Theorem may be stated and proved. As wasnoted inthe proof of the

Lemma,

thehypothesestoTheorem entail

(2.4)

^andsonecessarily b

n--oo.

^{However,it is not}

assumed that

{bn, n_> 1}

^ismonotone.

(In

^mostSLLN results,monotonicity of

{bn, n> 1}

^is

assumed.)

THEOREM 1. Let

{Y, Yn, n>l}

^bei.i.d, random variables and let

{an, n>l}

^and

{bn, n>l}

^be constants satisfyingan0,

bn>0

n>l, andeither

(2.2)

^or

(2.3).

^If

(2.1)

^{holds,then theWLLN}

n

(yj )

Eaj=l EYI(IYI-<Cn)

(3.1) bn -.P0

obtains.

PROOF.

P/]J=1" .

Define

^{ai(Y;-Ynj) -n} Ynj Y.iI(IYj >} ^I-<cn), <PJ[-Y.:f:Ynj]}<nP{[Y,>cn}=O(1)(by(2.1)). _{[j=t_.,}

^l_j_n,

^n_

^{I. For}arbitrary >0,

WEAK LAW OF LARGE NUMBERS FOR NORMED WEIGHTED SUMS 195 whence

Also,

sincefor arbitrary >0,

J= -.P0.

bn

j=l

bn

P

(3.2)

(3.3)

1.._[_

[

n

ai2Ey21(iYl<cn) o(1)

> ^_<

e2b2

nj =lJ

by theLemma. The conclusion

(3.1)

follows directly from

(3.2)

and

(3.3).

E!

REMARKS.

n 2

"

^a.

^O(na2n),

^then

j=l

(i) Apropos

of the condition

(2.2),

^if

cn/n

^is slowly varying at infinity and

or

n 2 j=l

j=l PROOF. Notethat

n 2

j=l

a. Cna2n ^C(n/cn)

²

_o(I) b2n

by slowvariation

(see,

^e.g.,Seneta

[8,

p.

18]).

Thenslowvariationyields

(see,

e.g., Feller

[9,

^p.

281])

o( 3 ^o

j=l

(ii)

^Adler

[10] ^provM

^a

rtial

^{converofTheorem1.}

(iii) In^thespiritofKl and Teicher

[5],

^Adler

[11]

hemploy Threm ^{to obtaina}generaliz one-sid law of the iterat logarithm

(LIL)

^{for weight sums ofi.i.d,} random variabl barely ^{with or} without finitemeanthereby generMizing^meofthe work of

[5].

(Corollary ^lowh

n

obtainby

KI

and Teicher

[5]

^{and they}

u

it in their invtigation of the

LIL

for i.i.d. ymmetric random variabl.) To ^mewhatmorescific, ^Adler

[11]

employed theWLLN

(3.1)

^{to obtain thea.c. limiting}

valueof me

(nonrandom)

^subquenceof n

Yj/b

ⁿ thereby yielding^anupr undfor thea.c.value of n

limninf

a.Y./b

n.

j=l

The ensuing Corollary is a WLLN analogue ^of Feller’s

[12]

^famous generalization of the Marcinkiewicz-ZygmundSLLN.

COROLLARY

(KI

and Teicher

[5]).

^Let

{Y, Yn, nl}

^i.i.d, random variabl and let

{bn, nl} itive

constants such thateither

bn/n

^or

2

bn’,

1, , ^bn

^and

1( )

^O

^(3.4)

Then n

E

^Y"

nEYl(IYl<bn)

j=l

bn

^{0 iff}

nP{lYl>bn} o(I).

196 ADLER

PROOF. Sufficiency followsdirectlyfrom Theorem whereasnecessityfollows fromthedegenerate criterion noting that the family

{(Yj- EYl(IYl__bn))/bn,

^l_<j_n,

n_l}

^{is uniformly}

convergence

asymptoticallynegligible. ^[]

REMARK.

In theKlass-Teicher

[5]

^{version of}Corollary 1, the second condition of theassumption

(3.4)

appears inthe stronger form

bn/n0.

Thenextcorollaryis an immediateconsequence of Corollary and istheclassicalWLLNattributed toFeller by Chow andTeicher

[6,

^p.

128].

COROLLARY2. If

{Y, Yn, n_> 1}

^{are i.i.d,}random variables, then n

j=IYj

^Vn P n -0 forsomechoiceof centering constants

{v

n,

n_> 1]

iff

nP{IYl>n} o(1).

Insuchacase,

vn/n EYI(IYI_<n + ^o(1).

Thenextcorollaryremovestheindicatorfunction from the expression in

(3. I).

COROLLARY3. Let

{Y, Yn, n_>l}

^{be i.i.d.}L randomvariablesand let

{an, n>_l}

^and

{bn, n>_l}

be constants satisfying

an:/:0 bn>0

n>l, andeither

(2.2)

^or

(2.3).

If

(2.1)

^{holdsandM lim} n

a./b

n

n--,oj=

existsandisfinite, then

n

J.’.-,l _.P _M(EY).

bn

PROOF. Firstobserve that

(3.1)obtains

by Theorem 1. Now

n--,oolim

^{Cn oosince}

a2n o(b2n)

^by

(2.4).

Then by theLebesguedominated convergencetheorem,

EYI(IYI_<Cn)--,

EY,whence

E

n

a;EYl(lYl_<Cn)

j=l

bn ^M(EY)

whichwhencombined with

(3.1)

yields the conclusion. []

4. A WLLN

WITH

RANDOM

INDICES.

Inthissection, Theorem is extended tothecaseofrandomindices

{Tn, n_> 1}.

Noassumptionsare

made regarding the joint distributions of

{Tn, n_ 1}

whose marginaldistributions areconstrainedsolely by

(4.1). Moreover,

^{it is not}assumed that the sequences

{Tn, n_l}

^and

{Yn, n_l}

^areindependent of each other. Itshould be noted that the condition

(4.1)

^isconsiderably weaker than

Tn/c n-.Pc

^{forsomeconstant}

c[0,oo).

THEOREM 2. Let

{Y, Yn, n_l}, {an, n_l},

and

{bn, n_l}

satisfy thehypotheses of Theorem and let

{Tn, n_l}

be positive integer-valued random variablesand l_an--.oo be constantssuch that for someA>O

and

b[,n] O(b[,n])

^if>1

^(4.2)

WEAK LAW OF LARGE NUMBERS FOR NORMED WEIGHTED SUMS 197 hold. Then

j-laj(Yj TII -= _b[an] ^{Ynj) _p}

^O.

Forarbitrary>0 and all largen,

t ^{b[an] >} cJ

Tn Tn

< P X

^a.Y.

^# __lajYnj

<_ P _t

_j=l

^U I11>[.

t-

^+o() (by (4.1))

=(! ⁺ o(,))A[Cln]P{iYi><[ttn]} ^+o(1)

=o(1) (by (2.1))

thereby establishing

(4.3).

Thus,tocomplete the proof,itonly needstobedemonstrated that

Tn a.(Y. ^EYnj )

J=lJ

^"J

P0.

(4.3)

(4.4)

Tothisend,forarbitrary >0 and all large n,

EYnj)I

p]j=l

t %5 ^> }

A. ADLER AND A. ROSALSKY

(by (4.1))

=P

_<k_<[=n]

max

= a.

^Y

ⁿ .-EYnj ^> cb[an] ^+o(1)

-< )+

[n] =

(by the Kolmogorov inequality)

if

O<A<I

ifA

> (by ^(4.2)

^{and Cn}

]’)

o(1)

(by^the

Lemma)

thereby establishing

(4.4)

^{andTheorem 2. []}

REMARKS.

(i)

The refereetothis papersokindly supplied the following example which shows that Theorem 2canfail if the norming sequence

{b[an], ^n_l)

^isreplaced by

{Tn, n_l}.

^Let

{Y, YH, n>l}

^be

i.i.d. Cauchy randomvariablesand let

an=l bn=n I+, Tn=n,

an=n, n>l

where >0. Then

(2.1)

^and

(2.3)

hold and trivially

Tn/an--Pl

and hence the conclusion to Theorem 2 obtains, but

[ aj Yj EYI(IYI <Clan] E

^Y" p

j=l

_Tn J-

^{-/. O.}

(ii)

The referee alsosuggested that the authors lookintothequestionastowhetherinTheorem2the normingsequence canbetakentobe

_{bTn’ n_l_}.

The ensuing corollary providesconditionsfor theanswer tobe affirmative. Itshould benoted that the pair of conditions

(4.1)

^and

(4.5)

isequivalenttothesingle condition

which isclearly weaker than

Tn/an P

cforsomeconstant 0<c<oo.

COROLLARY4. Let

{Y, Yn, n_>l}, {an, n_>l}, {bn, n_>l},

^and

{Cn, n_>l}

^satisfythehypotheses of Theorem 2 and suppose, additionally, that

bnT

^{and forsome}

At>0

that

WEAK LAW OF LARGE NUMBERS FOR NORMED WEIGHTED SUMS 199

and

pTn

^an

^< ,,} ^{o(1) (4.5)}

b[anl O(b[Atanl)

^if

^{A’<I (4.6)}

hold. Then

j=l

bT

n

PROOF. Inviewof Theorem 2,it suffices toshow that

b[anl/bTn.,

^isbounded in probability, that is, for all >0, thereexistsaconstantC<ooandanintegerNsuch that for all

n>_N

(4.7)

Tothisend, let >0. If

At_>l,

then lettingC=I,the monotonicity of

{bn, n_>l}

guarantees that

b[an] ^< Cb[A ,an], ^{n>_ (4.8)}

whereasifA

t<l,

^then

(4.6)

ensures

(4.8)

^{forsomeconstant}C<oo. Thus,

(4.8)

^{holds ineithercase. Thenfor} all large n,

<_ P{[b[an]>CbTn’] ^{[Tn > [A’an]]} +}

_< P{b[an]>Cb[Atan] ^{} + (by bn]’}

^and

^(4.5))

thereby establishing

(4.7)

andCorollary4.

(iii)

The ensuing example shows that, in general, Theorem 2 can fail if the norming sequence

{b[an], n_>l}

^is replaced by

{bTn n>l}.

^Let

{Y, Yn, n>_l}

^{be i.i.d,} random variables with Y having probability densityfunction

l[e,oo)(y),

-oo<y<oo

f(Y) ;21oCg

y

whereCisaconstantand let

an=l bn=n Tn:[q-6,

an=n,

n_>

^1.

Nowfor alln_>3,employing Theorem of Feller

[9,

^p.

281],

oo

nP{[Y[>n}

^nC

n

y210g

y

dy

(I+o(1))C

logn

o(1).

All of the hypotheses toTheorem 2aresatisfied and hence theconclusion to Theorem 2obtains.

Assume,

however,that

A. A. ROSALSKY

Tn

j=l

aTn ^_J=

vj- P0 (4.9)

prevails. Then

n j=l

n

EYI([Y[ <n2)

^P

ButbyCorollary 2,

n

j=l_n

EYl(lYl<n)

0.

whencevia subtraction

EYI(n<IY[_<n2) ^o(1).

^Butforn_>3,

EYl(n<[Y[<n2) / ⁿ²

_n

^{y log’y}

^{C dy}

^C(,og

^log

^n2-

^{log log}

ⁿ⁾

^{Clog 2,}

acontradiction. Thus,

(4.9)

must fail.

The last corollary of this section, Corollary5, is arandom indice versionofthe sufficiencyhalfof Corollary 2, andit isTheorem 5.2.6 of Chow andTeicher

[6,

^p.

131].

Corollary 5 followsimmediatelyfrom Corollary4by taking

an-1 ^bn-n,

^an--n,n>_l.

COROLLARY5. Let

{Y, Yn, n_>l}

be^i.i.d, randomvariablessuch that

nP{[Yl>n} o(1)

^{and let}

{Tn, n_> 1}

be positive integer-valued random variables such that

Tn -,

cforsomeconstant 0<c<oo.

Then

TR

j=l

T

_n

J_ EYI([YI<n)

0.

5. AN INTERESTING EXAMPLE.

Inthislast section,ageneralization^ofaclaicalexampleispresented. Asequence ofweighted^i.i.d.

random variables

{anYn, n>l}

^is shown, via Theorem 1, to obey ^a WLLN. On the other hand, the correspondingSLLNisshown tofail. Itshould be noted that

ElY

^{cw. Theclassicalexampleis}the special case6= andan 1.

EXAMPLE. Let

{Y, Yn, n_>l}

^bei.i.d,random variableswithYhaving probability densityfunction

C61yl)

⁶

^I(_

f(y) y2(lo _s oo,_e]U[e,oo)(y),

where0<6_(1andC₆isaconstant. Then foreverysequence ofconstants

{an,

n_)

1}

^with

E

n ^a.Y.

_(5.1)

J= _P

_0,

but

WEAK LAW OF LARGE NUMBERS FOR NORMED WEIGHTED SUMS 201

n n

j-1 j=l

lirasup -lira inf and, consequently,foranyconstant

cE(-oo,cx)

n P j=l

n

li-*moo nlan[

^{=c =0.}

PROOF. Set

bn’-nlanl

^{n>l. Then}

cn=n

n>l, and both

(2.2)

and

(2.3)

^{hold. Nowforall n_>3,} employing^{Theorem of}Feller

[9,

^p.

281],

nP{[Y[>n} 2nC6

^dy

^o(1),

n

y2(log y)b

(log

n)

andso

(5.1)

follows from Theorem since

EYI(IYI<_n)

0,

n>_

t.

Next,forarbitrary0<M<oo,

E

^{ooensuresthat}

oo

P Iynl

,whenceby theBorel-Cantellilemma

Plim

^sup

^{’Yn__J >} M}> P{l ^>

^M

i.o.(n)/

^1.

SinceMis arbitrary,

n.Y. ^n-1

I]=1

^j=l

oo lim_n--ocsup

:’-’

^{lim sup}_n--oo

nlanl

andso

<

lira sup

+

^{lim sup}

_(n-l)

^a.c.,

no

nlanl

^n--o

lan_l

implying

(5.2)

^viasymmetry and the Kolmogorov0-1law. E!

ACKNOWLEDGEMENT. The authors wish to thank the referee for supplying the first example and for posing the question concerningnormalizationby

_{bTn n_>l}_

^{whichwerepresentedin the}Remarks following the Proof ofTheorem2. Theauthorsare alsograteful to the referee forher/his very carefulreading of the manuscriptand forsomecomments whichenabled themto improvethe presentation ofthe paper.

REFERENCES

1.

LOIVE,

M. ProbabilityTheory, Vol.I,4th ed., Springer-Verlag, NewYork, 1977.

JAMISON,

B.; OREY,

S.and

PRUITT,

W. Convergenceof Weighted

Averages

of Independent Random Variables, Z. Wahrsch.^verw.Gebiete4

(1965),

40-44.

3.

ADLER,

A.andROSALSKY, A. OntheStrongLaw ofLargeNumbers for NormedWeightedSumsof I.I.D.Random Variables,StochasticAnal. Appl.

fi (1987),

467-483.

4.

ADLER,

A.and ROSALSKY, A.On theChow-Robbins "Fair"GamesProblem, Bull.Inst.Math.

Academia Sinica17

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