STRONG LAWS OF LARGE NUMBERS FOR WEIGHTED SUMS OF RANDOM ELEMENTS
IN NORMED LINEAR SPACES
ANDREADLER
Department
ofMathematics Illinois Institute ofTechnology Chicago, Illinois 60616U.S.A.ANDREW ROSALSKY
Department
ofStatisticsUniversityof Florida Gainesville,Florida32611 U.S.A.
ROBERT L. TAYLOR
Department
of Statistics Universityof Georgia Athens, Georgia 30602U.S.A.
(Received December 16, 1988 and in revised form February 20, 1989)
ABSTRACT.
Consider asequence of independentrandomelements{Vn,
n>
in a real separable normed linearspace
(assumedtobe a Banachspaceinmostof theresults),andsequencesof con- stants{a,,
n>
and{ha,
n with0< b, "["
oo.Sets of
conditions areprovidedfor{an(V EVn)
n>
toobeyageneralstrong law oflarge numbersof the formaj(Vj EVj)/b
n--> 0 almost certainly. Thehypotheses involve the distributions of thej=l
{V,,
n>
},thegrowthbehaviorsof{a
n>
and{bn,
n>
},and for some of the results imposeageometricconditiononX.Moreover,
Feller’sclassicalresultgeneralizingMarcinkiewiez-Zygmundstrong lawof largenumbers is showntohold for randomelementsin a real
separable
Rademacher typep(1 <
p<
2)Banachspace.KEY
WORDSAND PHRASES.
Real separableBanachspace,
independentrandom elements, normedweightedsums,stronglawof
largenumbers, almost certainconvergence,
stochastically dominatedrandom elements,Rademachertypep, Beck-convex normedlinearspace,
Schauderbasis.(uniformly) tight sequence.
1980
AMS SUBJECT CLASSIFICATION CODES.
Primary 60B12, 60F15;Seconda
00,.508 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
1.
INTRODUCTION.
Let (fl,
F, P)beaprobability spaceand letX be a tealseparable normedlinearspacewith norm. I. It
issupposedthatX
isequippedwith itsBorelo-algebra8. Thatis,8 is theo-algebrageneratedbythe class ofopensubsets of
X
determinedby I.II. A random
element, Vinx
isan /:-measurabletransformationfrom fltothemeasurablespace
(X,8). Theexpectedvalue ofV,
denotedEV,
isdefined tobe the Pettisintegral provided itexists. That is,VhasexpectedvalueEVE X
iff(EV) E(f(V))foreveryfX*
whereX*
denotes the (dual)spaceof all continuous linear functionals on X. The definitions ofindependenceandidenticallydistributed for random ele- mentsaresimilarto thoseinthe(real-valued)random variable case.Considera
sequence
ofindependentrandomelements{Vn,
n> },
allofwhoseexpected
values exist. Let
{a
n>
and {b n>
beconstantswith0<
b"1"
**. Then{an(V EVn),
n _> is saidto obeythegeneral strong law oflarge numbers (SLLN)withnorming constantsIb
n,n> 1}
ifthenormed weightedsumE aj(Vi EVj)/bn converges
almostcertainly tothe zero element in
X
(denoted by 0),and this will bewritten--,
0a.c. (1.1)bn
Herein, the main resultsfurnishconditions on X,onthe distributions of the
{Vn,
n> 1},
and onthe growthbehavior of the constants{a
n,n>
and{bn,
n >_ which ensurethat theSLLN
(1.1) obtains.In
mostof the resultsX isassumedtobe aBanachspace,and inmanyof the results{V
n>
is assumedtobe stochasticallydominatedbyarandom elementV
inthe sense that for someconstantD <
P{ IV,,I > t} < DP{ IDVI > t}, >
0,n>
1.(1.2)
Of course,(1.2)
isautomaticwithV V
andD
if the{V
n,n areindependent andidenti- cally distributed (i.i.d.)and even in this case theresultsare new.In
Section 3,SLLN’s
areestab- lished undergeometric
conditions on Xwhereas in Section4,SLLN’s
areestablishedwithout such conditions. TheSLLN
problemwasstudiedbyAdler andRosalsky [1, 2]in the random variable case, andsomeof those results will be extendedtothe random element casein thecurrent work.Taylor
[3] provided
acomprehensive
andunified treatment of results under whose conditionsanjVj
---> 0a.c. where{V
n,n >_ areindependent, mean zero random elements in a realsepa- rable normedlinearspace
and{an
j,< <
n, n> 1}
is atriangular array ofconstants. Someo: tbcargumentsinTaylor’s monograph utilizeda result ofRohatgi
[4]
whichwillnow be stated.(Rohatgi’swork generalized earlier work of Pruitt
[5].)
THEOREM
(Rohatgi [4]).Let {Xn,
nI]
beindependent,
meanzero random variables and letX
be anLp
random variable for somep >
1.Suppose
thatXn,
n> 1}
isstochastically
dom- inatedbyX
in thesense thatPI IXnl
>t} -< P{
IXl>
t], t>0,n>l.Let {anj <
j<
n, n beconstantssatisfying
lira anj 0 foreach> lanjl O(1),
andj=l
max
a, jl O(n-1/(1>-1)).
l<j_<n
Then
anjXj -->
0a.c.j--I
In
Theorems8 and9andCorollary 2 of the currentwork,versionsof some of the results(1.3)
pr6sented
inTaylor[3] willbe obtained underlessrestrictive conditions butonlyforthe case where anjaj/b
n,<
j<
n,n>
1,where{an,
n>
and{b,,
n>
areconstants. Theargumentswill notinvolveRohatgi’s theorem but, rather,willemployCorollary
below.Corollary plays
arole intheproofssimilartothe role thatRohatgi’stheoremplayedinestablishingthe counterparts presentedinTaylor[3]. Corollary has lessstringentconditions thanRohatgi’stheorem when anjaj/b,, <
n,n>
1. Specifically,ifthat choice of{an
j,< <
n, n satisfies(1.3),thenan O(n -u(p-l))
(1.4)bn
which is stronger than the condition
O(n
-/p)
(1.5)bn
of
Corollary
1. Thus,if{an
j,< <
n,n satisfiesbrian O(n a)
forsome 1/2<
t< I,
thentoinvokeRohatgi’stheoremrequiresthat(1.4) and themomentcondition
E IX
p<
hold where p> + __1 >
2, whereastoinvoke
Corollary
merely requires that (1.5) and the(weaker)moment 1condition
E IX
P<
hold where2>
p---"
_> 1.For
example,fori.i.d,random variablesIX
butnnot>
fromwithRohatgi’sEX1
0,theorem (which wouldthe classicalKo.lmogorov
requireSLLN EX
2< 00).Xj/n -
0 a.c.follows fromCorollary
A
SLLNfor normedweightedsumsof random elements in a real separable Banachspacehas beenproved byMikosch andNorvai’a
[6],buttheft result and thecurrentonesdo notentaileach other.510 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
Forrandomelementsin a realseparableBanachspace, the study ofthe
SLLN problem
dates backtothepioneeringwork by Mourier[7] (seealso Laha and Rohatgi[8, p. 4:52]orTaylor [3, p.72])wherein a directanalogue ofthe classicalKolmogorov
SLLN
was established.More
precisely, Mouriershowed thatffIV
n>
arei.i.d, randomelements in a realseparableBanachspaceand ifElIVtl <
*,,,then(Vj EVI)/n
--> 0a.c. (Anewproofof Mourier’sSLLNhasrecently beenj=l
discoveredby Cuestaand
Matran [9].) For
randomvariables,theKolmogorovSLLN
wasgeneral- izedbytheMarcinkiewicz-Zygmund SLLN (see, e.g.,Chow and Teicher [10, p. 122])which, in turn, was generalizedbyFeller[11].A
randomelement version of Feller’s result ispresentedin Theorem 4 below wherein it is assumed that the Banachspace
isof Rademachertypep(I<
p<
2).2.
PRELIMINARIES.
Some
definitions will be discussed andlemmaswillbepresentedpriortoestablishingthe main results.Let
{Yn,
n>
be a Bernoullisequence,that is,{Yn,
n arei.i.d,random variables withP{YI P{YI
-I 1/2. Let X bearealseparableBanachspaceand letX X
xXxX x
and defineC(X)
{v
n> 1} X**" Ynvn converges
inprobabilityn-1
Let <
p<
2. ThenX
is saidto beofRademachertyp_epifthere exists aconstant0< C <
such thatEl
l, Y. nvnlIP_<CEIIVnllP
n=l n=l
for all
{vn,
n>
C(X).Hoffmann-Jdrgensen
and Pisier[12]
provedfor<
p 2 thatareal separableBanachspaceisofRademachertypepiffthere exists a constant0< C <
such thatEl
z_,Vjl It’ < CT_.,
ElIVil It’
foreveryfinite collection
{Vt
V ofindependentrandomelementswithEVj
0,El
IVjl It < ,,,,
<jn.Ifa realseparableBanachspaceis ofRademachertype
p
for some<
p<
2,then it is of Rademachertypeqfor all<
q<
p.Every
realseparableBanachspaceisofRademachertype (at least) whiletheLp-spaces
and-spaces
are of Rademachertypemin 2,p forp 1.Every
real separableHilbertspaceand real separablefmite-dirnensional.Banach space isof Rademacher type 2.A
normed linearspace
X is saidtobe Beck-convexif there exists aninteger N>
and a number0<I< such that for all choices of{v
vN}
X withlvjll
< 1, < <N,
I1+v
__.
:!:VNI
< N(1-E)for somechoiceof+and signs. This propertyhas beenextensively studiedby Giesy [13].
A
real separableBanach spaceisBeck convex iff it is ofRademacher type p for some p >1.A
_Schauder basis for a normed linearspace
is asequence {1
i,> 1}
c g suchthat foreach vX
thereexists aunique sequence ofscalars{t
i,1}
suchthatrn
lim
tii=v.
(2.1)rn--***i--I
A
sequenceof linear functionals{f
i,>
(calledcoordinatefunctional$for the basis{i, > })
canbedefinedbyletting
fi(v)
ti, :> 1, wherev X and(2.1) holds,andasequenceoflinear functions{Urn
m 2 (called partialsum operatorforthe basis{i, > })
can be definedbyUrn(v)- Z fi
(v)i,
i=l
The residual operators
{Qrn,
m>
aredefinedbyvX,m_>
1.Qm(v)
vUm(V),
v,
m>
1.A
Schauder basis is saidtobe amonotone basis
ifIUm(v) I,
m>
isamonotone sequencefor each v X.A
sequenceof random elements{V
n,n> 1}
in a normed linearspace X
is saidtobe(uni- formly) tight if for eachE>
0,there exists a compact subsetK
eofX
such thatP{V Kt}
2 Eforalln
>
1.LEMMA
(AdlerandRosalsky [1]). LetX
oandX
be random variablessuch thatX
o issto-chastically dominated by
X
inthesense that there exists aconstantD <
such thatThen for allp
>
0P{
IXol > t} <
DP{DX >
t}, t>O. (2.2)EIXolrrI(IXol <
t)< DtPP IDXI > t} + Dp+IEIXIPI(IDXI <
t),t>O.
(2.3)LEMMA
2 (Adler and Rosalsky [1]).Let IX
n> 1}
andX
be random variables such that{X
n> 1}
isstochasticallydominatedbyX
in the sense that there exists aconstantD <
such thatP{ IXnl > t} DP{ IDXI >
t},>
0,n 1.Let
{ca,
n1}
bepositiveconstantssuchthat[nax
Jcff]
jn---1 cff
O(n)forsomep>
0andP{IXI >
Den}
<oo. Thenforall0< M <-0, n=l512 Ao ADLER, A. ROSALSKY AND R.L. TAYLOR
----" EIXnIP I(IXnl <
Mca)
<n=l
Cn
pLEMMA
3.Let X
o andX
be random variables suchthatX
oisstochastically dominatedbyX
inthe sense that(2.2) holds. Thenand
EIXolI(IXol
>x)f P{ IXol
>t}dt+ xP{ IXol
>x}, x>
0 (2.4)EIXolI(IXol >
x) _<D2EIXII(IDXI >
x), x>
0.(2.5)
PROOF. Integration byparts yields(2.4), andthen(2.5) followsimmediately from (2.4)and (2.2).VlLEMMA
4(AdlerandRosalsky[2]). Let X
be a random variable such thatP{ IXI
> t} isreg- ularly varyingwith exponentp
<-1. ThenX Lp
for all0<
p <-p
andEIXII(IXI
>t)=(l+o(1))P+I tPIlXI >t}
ast--->,,,,.Thenextlemma shows that stochastic dominance canbe accomplished byasequenceof ran- dom variableshavingabounded absolutep-thmoment(p> 1).
LEMMA
5 (Taylor [3, p. 123]). Let {X n_> 1} be random variables such thatsu
n>_
ElXnl
P<
for somep>
1. Then there exists a random variableX
withE
IxIq<
for all 0 < q < psuch thatPI IXnl
>t}
<P{
IXl > t},>
0, n > 1.Finally,aremark about notation is in order.Throughout,the symbol
C
denotes a generic con- stant(0< C <
*,,)which isnotnecessarilythe same one in eachappearance.3.
SLLN’S UNDER PROBABILISTIC AND GEOMETRIC CONDITIONS.
Withthesepreliminaries accounted for,the firstgroupofresultsmaybe established. The ran- dom elementsareassumedtobeindependent, andgeometric conditions are
placed
on the realsepa- rablenormed linearspace. Thespace
isassumedtobe a BanachspaceofRademachertypep (for suitablep)in Theorems 1-7,anditisassumedtobe Beck-convex ha Theorem 8. Thenextlemmais thekey lemma inestablishingTheorems1-4.LEMMA
6.Let {V
n,n> 1}
be independent random elementsin a realseparable,
Rademacher typep (1<
p< 2)
Banachspace
X.Suppose
thatIV
n,n>- 1}
isstochasticallydominatedbya ran-dom elementVinthe sense that(1.2)holds. Let
{a
n and{bn,
n:> beconstantssatisfy- ing0<
bT
and(3.1)
x-"z_,p{ lanV] > Dbn} <
0% (3.2)then
Zaj(Vj- EVjI(I lajVjl < D2bj))
j=l
--
0a.c.(3.3)
PROOF. Let
Cn- anl Yn VnI(llVnll < D2cn),
n _> 1. (3.4)Thenfor n
>
aj(Yj- EYj)
[I p
(since
X
is ofRademachertypep) o(1) (byLcmma
2),whence
EII -SI IP--
0=I bj
for some random clementSin
X
implyingn
X aj(Yj EYj) p_. S.
j=
bj
Sinceconvergence inprobability and a.c.convergencearccquivalcntfor sums ofindcpcndcntran- domelements in aseparableBanach
space
(seeIt6 and Nisio[14]),., aj(Yj- EYj)
implyingvia theKronecker lemrna that
converges a.c.
X EV)
b
--
0a.c. (3.5)514 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
However,
P{liminfIV Yn]}
bythe Borel-Cantelli lemma since(1.2) and(3.2)ensure thatP{Vn* Yn} P{llVnll > D2cn} < D ,
P{IIVIi> Dcn}
<,.
n=l n=l n=l
The conclusion(3.3)then followsdirectlyfrom(3.5).
D
In
the firsttheorem,there is atrade-offbetween theRa(lemachertype and the condition(3.6);thelargertheRademachertypep,the condition(3.6)becomes less stringent (since
bn/la ’).
THEOREM 1. Let
{Vn,
n2 beindependent random elementsin arealseparable,Rademacher typep (1 _<p < 2)Banach
space. Suppose
that{Vn,
n>
1} isstochastically dominated by arandomelementVinthe sense that(1.2)holds. Let{an,
n>
and{b
n bcconstants satisfying 0<
bT ,,,,, bn/la T,
b -lajlP
-I an I’P jn’-jP
O(n)’ (3.6)and
lanl j__l’j
O(n)"If theseriesof(3.2) converges,then theSLLN
(3.7)
obtains.
Z aj(Vj- EVj)
-->0a.c.
PROOF.
Define {c n>
1} and{Yn,
n>
1} asin(3.4). Note attheoutsetthat(3.7)ensures that c< Cn,
n>
1, and so for all>
1,by (1.2)and(3.2)
, PII IVjl
>CD2n} < D , PII IVI
>CDn}
n=l n=l
< D
P{I
IVl> DCn} < ,,,,.
n--’l
Thus, Ell
Vjll
<,,
>_ 1,and so (see, e.g., Taylor [3, p.40]) Vj, >
1, all haveexpectedvalues.Also, c
T
by(3.6).Next,
(3.3)holdsby Lemma6 and soitonly needstobedemonstrated thatajEVjI(I]Vjl] > D2cj)
b
Tothisend,
El
IVnl
II(IIVnll
>D2cn)
n=l
Cn
<_D2 _1
EIIVIII(ilVII >Dcn)
(by (2.5))n=l Cn
D2 .1
ElIVI II(Dcj
<IVII
<Dcj+ 1)
n=l
Cn
j---nj+l
<
D
2 ElIVI II(Dcj
<IVII
<Dcj+l)
n=l
Cn
<
D3 cj+lPIDc
< lVll< Dcj+)
(by(3.7))
J=- cj+
< C jPIDcj <
lVll< Dcj+}
j=l
C PIDcj <
lVll< DCj+l}
j=l n=l
C E E PIDcj <
lVll< Dcj+l}
n=l
C P{IIVII > Den}
<** (by(3.2)),n=l
whencebytheKronecker lemma
I
ajEVjI(I IVjll > D2cj)l lajlEI IVjl
II(IIVjll > D2cj)
j-I j-I
o(1).l-I
REMARK. Apropos
of Theorem1,the authors are abletoshowthrougha slight modification ofthe argument thattheconditionbn/lanl T
can bereplaced
bythesomewhat weakerconditionbn/la
n O(infbj/l
ajI).
THEOREM
2.Let {Vn,
n>
be independent random elementsin a realseparable,
Rademacher type p
(1 < p <
2) Banachspace. Suppose
that{Vn,
n> I}
isstochasticallydominated byarandom elementV
inthe sense that(1.2)
holds,andsuppose
thatElIVll< .,,. Let
la
n,n>
andIb
n,n>
beconstantssatisfying 0<
bn’1"
**,(3.1),andajl O(bn).
j=l
If theseriesof
(3.2)
converges,then theSLLN
(3.8)
obtains.
bn --)0a.c.
516 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
PROOF.
Define{c
n,n>
1} and{Yn,
n> 1}
as in(3.4). Note
attheoutsetthat(1.2) guaran- teesthatEllVnll
<,
n 1, andsoV
n,n 1,all haveexpected
values.Now (3.3)
holdsbyLemma
6and so itonlyneedstobedemonstrated thataEVI( V > D2c)
j=l --0.
bn
To
inatedthisconvergence
end,notethattheorem(3.1)
ensures c--
-0,whence by(2.5),
ElIVll<
**, andthe Lebesguedom-II EVaI(I
VII > D2Cn)l
EllV II
I(ll V >D2Cn)
g
D2EI IVI II(I
IVll>
Dcn) o(1).
But
thenby(3.8)
andtheToeplitzlemmaII ajEVjI(IIVjll
>D2c)II lajl lEVjI(IIVjll >D2cj)II
bn
bn o(). r
THEOREM3. Let
{V
n beindependent randomelements in arealseparable,
Rademacher typep (1
<
p< 2)
BanachslSacc. Suppose
that{V
n,n>
is stochastically dominated byarandomelement Vinthe sense that(1.2) holds, andsuppose
thatP{I
IVl> t}
isregularly varyingwith exponentp <
-1. (3.9)Let {a
n>
and{b
n,n>
1 beconstantssatisfying 0<
b’1"
and(3.1). Iftheseries of(3.2)converges,
then theSLLN
obtains.
b
--)0a.c.
PROOF. Define
{c
n,n> 1}
and{Yn,
n> 1}
asin(3.4).
Now ElIVII
< by Lemma4 and so(1.2)
ensuresthat EllVal <
00,n>
1, implyingthatn,
n>
1,all haveexpectedvalues. Again (3.3)holdsby Lemma 6and so itonlyneedstobe demonstrated thatajEVjI([ Vjl
>D2cj)
J=l --)0.
ba
To
thisend, itfollows from(2.5), (3.1),andLemma4 that for all n>
some no ElIVnl
II(IIVnll > D2cn) < D2EI IVI
II(I IVll>
Dcn)
< CcnP{ IVI > Den}.
Thenby (3.2),
and so
----1
ElIVnl
II(IIVnll
>D2cn)
gC + C PIIIVII
> Dcn)
<n=l Cn n
I1 ajEVjI(I IVjll
>D2cj)ll lajlEI IVjl
II(IIVjll
>D2cj)
j=l S j=t o(1)
bn bn
by the Kronecker lemma.I"1
REMARK. Apropos
ofTheorems 1,2,or3,Example
of Adler andRosalsky [2]
shows that the Theorems canfail withouttheassumption
(3.7), (3.8),or(3.9), respectively.Theensuinglemma can be
helpful
inverifying the conditions(3.6), (3.1), (4.6)
ofTheorems 1, 2, 3,or 11, andit willbe usedintheproofof Theorem 4.LEMMA
7(Adler
andRosalsky
[1]).Let {Cn,
n> 1}
beconstantswith0< cnP/n "1"
for some p>
0. Theniff
lim inf
cpm
>
r for someintegerr>
2.tl---
cP
Thenexttheorem is a random element version ofaclassicalresultof Feller[11 whichhad extended theMarcinkiewicz-Zygrnund SLLNtomoregeneral normingconstants.
THEOREM
4. Let{Va,
n bei.i.d,random elements inarealseparable,Rademacher type p (1 < p<
2)Banachspaceandlet {bn,n> 1}
bepositiveconstants.Suppose
that eitherbn bn
(i)
EVI=0, ,1,, ---’1’
for somex>--n na p
or
(ii)
E V)
**,"I’.
n If
P{IIVIII
>bn}
<*,,, (3.10)then
j=l
bn --*
0 a.c. (3.11)518 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
PROOF. In
either case b"r
andblain ’.
Nowbn/nI "
where otin case (i) andI
incase(ii). Thus,
and soby Lemma 7
b’n (2n)lP
2Ip 2,liminf--- >liminf
>
n-oo-
bnP
n--nlP
Thenbyl..emma6
bnP Z
O(n).(Vj EVjI(I IVjll
_<hi))
j=l
Oa.c.
bn
In
case(i),bn/n ,l,
and(3.10)
entail(scc ChowandTichcr [10,pp.
123-124])(3.12)
j=l
<
j=-I o(I)b
bn
which when
combined
with(3.12)
yields(3.11)
sinceEV
0.In
case(ii), in view of(3.10),
necessarilybn/n ’l"
and so(seeChow and Teicher[10, pp.
123-1241)
II EVjI(I IVjll < bj)ll
j=l
b.
yielding (3.11)via(3.12).v!
o(1)
REMARK. In
the special case whereEV
0,EI VI
q< for some<
q<
p<
2,andb n
l/q,
n>
1,Theorem4(i)reducestothe Marcinkiewicz-Zygmund typeSLLN
Vj/nI/q 0a.c.of Woycz’yfiski
[15]. Woyczyfiski’s
resulthasbeen improvedby
deAcosta [16].
Forsomerelated results, see
Wang
andBhaskaraRao 17].THEOREM
5.Let {V
n,n> 1}
beindependentrandom elements in a real separable, Rademacher typep (1<
p < 2)Banachspaceandsupposethatsu"
ElIVnl IP
Let {a n_> 1} and
{bn,
n_> be constantssuch that0<b’I"
andThen theSLLN
an O(n-I/P(log
n)-I/q)
for some 0< q< p.bn
(3.13)
(3.14)
obtains.
aj(Vj- EVj)
j=l
brl
---) 0a.c. (3.15)PROOF.
Condition(3.13)
ensures thatVn,n>
1, all haveexpectedvalues. Letcbn/lanl,
Yn VnI(I IVnll < Cn),
n> 1. Now by (3.13)and(3.14)El
IYnl
p ElIVnl IP
1Z -<Z _<cz
<n=
c.r’
.=,c
.=,c
implying (see theproofofLemma 6)
(3.16)
Now
aj(Yj- EYj)
j=l
b ---) 0 a.c.
(3.17)
El
IVnl IP
)",P{V,
#Y,} PIIIV, > %1 < < (3.18)
n=l n=l n=l CnP
recalling (3.16), whence by theBorel-Cantellilemma P{liminf
[V
nYn]}
implyingvia(3.17)
that
aj(Vj- EYj)
j=l 0a.c. (3.19)
Next,
Z __1 EliVnllI(lIV,all >ca)
n=l
Cn
Y’P{ ’Vnl’ > Cn} + X "n] P{I ’Vnl’ >
t}dtn--I
Cn IP
(by(3.18))
<c+cz
1n=l
CnP
(by(3.13)
and(3.16)),and sobythe Kronecker lemma
(by
(2.4))
1 ajEVjI(lIV1ll > cj)
llb yielding(3.15)via
(3.19).
E!lalEI IVjl
II(IIVjll
>cj)
b o(1)
THEOREM6.
Let {V
n>
be independent random elements inarealseparable,Rademacher typep (1
<
p 2)Banachspace.Suppose
that{Vn,
n>
1} isstochasticallydominated by arandom elementVinthe sense that (1.2)holds, ands.uppose
thatEllWllq< forsome520 A. ADLER, A. ROSALSKY AND R.L. TAYLOR _<q < p. Let
{a
n>
and {b n >_ beconstantssatisfying0<b" ,,
(3.8),andThentheS
LLN
an O(n-l/q).
bn
obtains,
ba
(3.20)
--) 0a.c.
(3.21)
PROOF. Note
that(1.2)entailsE lIVnJlq
<0% n>
1, and henceV n>
1, allhaveexpected values. Letcbn/I
aI, Yn VnI(I IV < nl/q),
n 1.Now
El
IYnl IP
np/q nl/q<D P{IIDVII >
.= c .= c
4-
Dp+ _1
ElIVI lPI(I
IDVll n/q)
(by (2.3))-1=
C/
C n’-q
ElIVI IPI((k-1)
uq<
lDVll kuq)
n=l 1-1
(by (3.20)andEll V Iq<,,o)
C + C)’
El IVlIPI((k-1) vq < IDVII <
kl/q)
n"-qk--1 n----k
<
C+
C k(q-p)/qElIVIIPI((k-l)
l/q<IDVI <
k/q)
k-=-I
<
C+
CEI IVI
IqI((k-l)uq < IDVII <
kuq)
k=l
C + CEI IVI
Iq<
implying (see the
proof
ofLemma 6)
ajCgj- EYj)
bn
--
0 a.c.Now
by (1.2)andElIVl q<
0%PlVn Yn} P{ IVnl > nl/q} < D P{ IDVI >
nl/q} <
*%n=l n=l n=l
and sobythe Borel-Cantelli lemmaP{liminf
[V Yn]}
implyingvia(3.22)
that(3.22)
aj(Vj- EYj)
j=-l
b
---) 0a.c.
Next, by (2.5),
EIIVII < ,,,
and theLebesguedominatedconvergence theorem EllVnl
II(IIVnll
>nI/q) < D2EI
IVlII(IIDVII
> nI/q)
o(I),whenceby
(3.8)
andtheToeplitzlernrna(3.23)
bn bn
yielding
(3.21)
via(3.23).laIEl IVjl
II(lIVjll > jl/q)
<
j=l o(I)Thefollowing
Corollary
isan extensionofTheorem2of Adler andRosalsky [2]
(which,in tum, is an extension ofTheorem3.1 ofFernholz and Teicher[18])and establishesaSLLN for normed weighted sums ofstochasticallydominated random variables. Itwill be used in theproofs
of Theorems8 and9butmaybe ofindependent interest.COROLLARY
1. Let{X
n> 1}
beindependentrandom variables and letX
be anL
vran- dom variableforsome<
p<
2.Suppose
thatIX
n> 1}
is stochastically dominatedbyX
inthe sense that there exists aconstantD
< suchthatP{ IXnl
>t} < DP{ IDXI
> t},>
0,n>
1.Let
{a
n>
and{bn,
n>
beconstantssatisfying0<
bT
*,,,an/b O(n-VP),
and(3.8).Then theSLLN
obtains.
b --)0 a.c.
PROOF. Since(R, I’!)isa realseparable,Rademacher type 2 Banachspace,theCorollary fol- lowsimmediatelyfrom Theorem6withp 2andq
p <
2.!"1THEOREM 7.
Let {V
n> 1}
beindependentrandom elements in a realseparable,
Rademacher typep (1 <p < 2)Banachspace.
Suppose
thatIV
n> 1} isstochastically dominated byarandom elementV
inthe sense that(1.2)holds, andsupposethatE IVI IP
<**. Letla
n>
andIbn,
n>
beconstantssatisfying 0<
b"1"
,o,(3.8),and(3.14). Then theSLLNobtains.
0a.c.
PROOF. Using the truncation
Yn VnI(I IVn
Snl/r’),
n>
1, theargumentisaslightmodification of that usedtoestablishTheorem6. The details arelefttothe reader. 121
REMARK. An
interestingquestionwhich weareunabletoresolveiswhether Theorem 7 holdswith(3.14) replaced bythe somewhat weaker conditionan/b O(n-UP). Moreover,
Theorem522 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
7 should becomparedwithTheorem I0 wherein the
{V
n> I}
are(uniformly)tight.ThenextTheorem establishes aSLLNfornormed weightedsumsof random elements in a real separablenormed linearspacewhich isBeck-convex.
It
shouldbecompared
withTheorem5 of TaylorandPadgett[19]
(orTheorem5.3.1 ofTaylor [3, p.137]).
THEOREM
8. Let{V
n,n> 1}
beindependentrandomelementsin a realseparablenormed linearspacewhich is Beck-convex and let {a n>
1} andlb,,
n>
1} beconstantssatisfying an>0,n>l,0<b nT**, aj=O(bn),an/b n= O(n -l/p)
forsome<p<2,
andj=l
_(aj
dn)
o(bn)
(3.24)whered
n--
min aj,n>
IfE IVnllq
l<j
,
< for someq > p,then theSLLNobtains.
b
--->
0 a.c. (3.25)PROOF.
Without lossof generality,itmayandwillbesupposedthatEV 0,n>
1.Sup-
pose,initially, that the{V
n,n>
areuniformly bounded in the normby aconstant, that is,lV.[[ < C
a.c. Then, sincenda <
aO(b.),
j-I jl
b. b.
@.b.
cz %-%)
+
0a.c.bn n
by
(3.24)
d aSLLN
ofBeck [20,eorem 10]
(which iseorem
4.3.1 ofTaylor[3,
p.87]) thereby pvg
theeorem
when( IV
1C
a.c.Next,
in general, defineX.=VnI(IIV.II
<M),Yn=V.I(IIV.II
>M), n21,where
< M <
is aconstant.By
theportionof the theoremalready proved,
Note
thatforn 1,aj(X- EXj)
b ---)0 a.c. (3.26)
E{Mq-IIVnlII(IIVnil >M)}
El
IYnll Mq-I
El
IVnl
IqI(IIVnll >
M)C
<
<Mq_ Mq_
and so in view of aj O(b
n)
j=l
<
j=l+
j=-Ib
bn bn
j=l
+
bn
X aj(I IYjll
ElIYjl
I)<
j=l+
bn
2
Z ajEI IYjll
b
C Mq-I
Now
{I Ynll E lIYn II,
n> I}
areindependentmean0 random variableswith su ElIYnl
lq<
2q ElIVnl
lq,l E IIYnll EllYnll q<2q n>l na
By
Lemma 5,thereexists arandom variableY
withElY p<
such thatPtlIIYn’I -EllYnll l>t} <P{IYI >t}, t>O,n>
1, whenceby Corollaryb --> 0a.c.
But
thenby(3.26)
and(3.27)
(3.27)
<
limsupn- b
I, aj(Yj- EYj),I +
lknsup j=ln-
b.
S
C
a.c.Mq-I
and since
M
isarbitrary, the conclusion(3.25)
follows.[]4.
SLLN’S UNDER PROBABILISTIC CONDITIONS.
In
this section,SLLN’s
are obtainedwithoutimposinggeometric conditions on the Banach space.As
in Section3,momentconditions areplacedonindependentrandom elements andrestric- tionsareplacedontheconstants{an,
n>
and{bn,
n>
}. InTheorem9,the Banach spaceis assumedtoadmit aSchauder basis andinTheorem 10,theindependentrandomelementsin a524 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
Banachspaceareassumedtobe (uniformly) tight.
ForaBanachspaceadmittingaSchauder basis, recall the definitions of
If
i,>
],U
m,m> 1},
andQm,
m> 1} presented
in Section 2. Theorem 9 should be comparedwith Theorem5.1.4ofTaylor [3,p.114].
THEOREM
9.Let {V
n,n>
beindependent,mean zero random elements in a realseparable Banachspace admittingaSchauder basis{13
i, > 1}.Let
{a n>
1} and bn,n> 1}
beconstants satisfying 0<
b"
*,,,(3.8),andan O(n -I/p) (4.1)
bn
for some
<
p<
2.Suppose
that there exist randomvariables{Xi,
and{Ym,
m2 and a constantD<
such thatPllfi(Vn) > t} DPIIDXil > t}, >
0,nR 1,> 1, P{ 111Qm(Vn)l
ElIQm(Vn>l > t} <
DP{IDYml > t},
su
t
ElX il
P<*%supEIYmlP<,%
m’l
and
t>0,
m
l,n> 1,Then the
SLLN
lim
.
ElIQm(Vn)l O.
(4.2)-
0 a.c.obtains.
PROOF.
Itfollows directly
fromCorollary
thatbll
--)0 a.c. for eachand
lal(I IQm(Vj)I
ElIQm(Vj)I
I)Tin,n_=
j--1 0 a.c. for eachm>
1.Then
i-I
b
j--I
b 1113ill
0 a.c. for eachm> 1.(4.3)
(4.4)
Thus,by (4.4), (4.3), (3.8),and(4.2)
J=Ibn < lUm t’-bn II+
II EajQm(Vj)
j=l
b
[ b" II + T= + C
ElIQ=(Vj)I
---,
0a.c. as firstn-***and thenm-***,v!THEOREM
10.Let {V
n,n> 1}
be a(uniformly) tightsequence
of independent,meanzero randomelementsin arealseparable
Etanachspace X. Let {a
n and{b
n>
beconstants satisfying 0<
bn’l"
**,(3.8),and(4.1)
for some 1<
p<
2.Suppose
thatIV
n,n:>1}
isstochasti-callydominatedbyarandomelement
V
inthe sense that(1.2)
holds,andsuppose
thatE IV
liP<
**. Then theSLLNobtains.
0 a.c.
PROOF. Let
hbe anorm-preserving, bicontinuous,linearmapping ofX
intoC[0,1]
(--the Banachspaceof all continuousreal-valuedfunctionsyon[0,1]
withnormII
y llul
ly(t) l).TheBanach
space
C[0,1]admits amonotonebasiswhereIQm(y)l lyll
andIfm(y)l < lyll
for eachy [0,1]
andm>
andwhereIQm(y)l I,
m 1 is amonotonedecreasingsequence
for eachy C[0,1].
Then{h(Vn),
n11
isa(uniformly) tightsequenceofindependent,mean zero randomelementsinC[0,1]. Now
for arbitraryI>
0, chooseu>
0 sothatD2EII V
llI(11Vll>
u)< -.
ThenLemma 3 provides EllVnl II(I IV, II >
u)<
-
for all n 1.By
(urfiform) tightness,acompactsbsetK
ofC[0,1] ma,y
bechosensothatPIh(V,)
dKI < 3"-’"
for all n 1,whenceEl
IVnl II(I IVnll
_<u)I(h(Vn) a
K)S for all n>
1. SinceIQm(y)l
foreachyinthe compactsetK,
Dini’stheoremensures thatthere exists anintegermosuchthatsup
Q=(y) < e
yK frall m
> rn"
Thenfrall m> rn
and n>
EI Q=(h(Vn))l < EI Qm(h(Vn))I(I IV
Su)I(h(Vn) K)
+ElIV
nllI(llV nIl <u)I(h(V,)aK)+EIIVnllI(llV"ll
>u)<526 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
therebyestablishing(4.2) for the randomelements
Ih(Vn),
n> }.
TheidentificationsX
IVl andYm IVI + DE
IIVll for all>
and m>
ensure that the other conditionsofTheorem9 hold. ThusZajVj h(ZajVj) Zajh(Vj)
j-I j=l j--I
bn bn bn
--->0 a.c.V1REMARKS.
(i)When an 1,bn n, n>
1,and EllV II
P<
for somep>
1, Theorem 10inconjunctionwithLemma
5will establishtheSLLN ofTaylor [3, Corollary 5.2.9, p.133]. As
pointedoutby Taylor [3,p. 133],
that sameSLLN
canbe obtained from Theorem5.2.8ofTaylor [3, p. 13I]butunder the stronger assumption thats
EllVal
P<
for somep>
2.(ii)Theorem 10canfail ifp andEl IVll
,,,,.
Foranexample,seeTaylor [3, Example5.2.3, p. 135].ThenextCorollary should becomparedwithTheorem5.2.8 ofTaylor
[3,
p. 131 ].COROLLARY
2.Let {V
n,n >_I}
bea(uniformly) tightsequenceofindependent,mean zero randomelements in a realseparableBanachspace.Let {an,
n> 1}
and{b
n> 1}
be constants satisfying0<
b’1"
**,(3.8),and(4.1)for some<
p<
2. Ifthenthe
SLLN
Ell
Vnll
q<
forsomeq > p, (4.5)
obtains.
ZaV
b 0 a.c.
PROOF.
Condition(4.5)ensuresbyLemma
5that(1.2)obtainsandEl IVl p< .
TheCorollarythenfollows from Theorem 10. 121
In
thenextCorollary,thesequence {Va,n :>1}
is i.i.d, and themomentcondition(4.5)is weakenedtoEllVIIIP.
TheCorollaryshould becomparedwithTheorem5.1.3ofTaylor[3,
p. 112].
COROLLARY 3. Let
{V
n>
be i.i.d,mean zero random elements inarealseparable Banachspace. Let{an,
n> 1}
and {bn,n> I}
beconstantssatisfying 0<
b"1"
*,,,(3.8), and(4.1) forsome<
p <2.If ElIVll IP
<,,
then theSLLNobtains.
bn
---) 0a.c.PROOF. Since thei.i.d,hypothesisensures that
{V
n> 1}
is automatically (uniformly)tight (see Taylor[3, p. 121]),theCorollary follows immediately from Theorem 10.[]REMARKS.
(i)In
theparticularcase where a 1, b---
n, andp 1, Corollary 3reducesto theSLLNofMourier[7].(ii)
A
Fr6chetspace_is acompletelinear metricspace. Using Theorem 10,aSLLN
maybeobtained for random elements in arealseparableFr6chetspaceF
which is alocallyconvexspacewitha countable family of seminorms{Pk,
k>
1} defined on itsuchthatthemetricdis definedbyd(x,y)
Pk(x Y)
k=l
2k(1 +
pk(X y))forx,y F.
Thedetails will notbe givensincethe argumentparallels thatof Theorem5.2.10ofTaylor [3, p.
136]. (Corollary2plays the same role in theproofasTheorem5.2.8 ofTaylor [3,p. 131] played in provingTheorem5.2.10.) Infact, almost all of the resultsirt this sectionhaveparallelresults for Fr6chetspaces.
Inthelast theorem, there is noindependence assumption onthe sequenceof random elements.
Moreover,
the spaceis equippedwithaseminormp
whichis notnecessarily a norm and thus the resultisapplicabletoalargerclass ofspacesthanrealseparable normedlinearspaces."Eae
definitionofrandom element isanalogous
tothat discussed in Section forrealseparable
normed linearspaces.THEOREM
11. Let{Vn,
n> 1}
be random elements in a realseparableseminormedlinearspace
with seminormp. Suppose
that{Vn,
n> 1}
isstochasticallydominatedbyarandomelementV
inthe sense thatthere exists aconstantD <
such thatP{p(V n) > t} < DPIp(DV) > t}, >
0,n>
1.Let an,
n> 1}
and {b n> 1}
beconstantssuch that0<
bn"1"
andSjSn
lajl Jj-’gTJ =O(n).
(4.6)528 A. ADLER, A. ROSALSKY AND R.L. TAYLOR
hen
P{p(anV) >
Dbn} <
*, n--1bn
PROOF. Set Yn P(Vn),
n _> 1,andY p(V).
Thenby Theorem 2 of AdlerandRosalsky [1],p ajVj ajl p(Vj)
:
j--1bn bn
---,
0 a.c.r"lREFERENCES
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..ErgodicTheory; ProceedingsofanInter- nationalSymposiumHeldatTulane University,New
Orleans,Louisiana,October,1961
(Ed.F.B. Wright),21-53, Academic
Press, New
York, 1963.Boundary Value Problems
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