R E S E A R C H
Open Access
Complete moment convergence for
randomly weighted sums of martingale
differences
Wenzhi Yang
1, Yiwei Wang
2, Xinghui Wang
1and Shuhe Hu
1**Correspondence: hushuhe@263.net
1School of Mathematical Science, Anhui University, Hefei, China Full list of author information is available at the end of the article
Abstract
In this article, we obtain the complete moment convergence for randomly weighted sums of martingale differences. Our results generalize the corresponding ones for the nonweighted sums of martingale differences to the case of randomly weighted sums of martingale differences.
MSC: 60G50; 60F15
Keywords: complete convergence; randomly weighted sums; martingale
differences
1 Introduction
The concept of complete convergence was introduced by Hsu and Robbins [],i.e., a se-quence of random variables{Xn,n≥}is said toconverge completelyto a constantCif
∞
n=P(|Xn–C| ≥ε) <∞for allε> . In view of Borel-Cantelli lemma, this implies that
Xn→Calmost surely (a.s.). The converse is true if{Xn,n≥}is independent. Hsu and
Robbins [] obtained that the sequence of arithmetic means of independent and identi-cally distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Erdös [] proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory, and it has been generalized and extended in several directions by many authors. Baum and Katz [] gave the following generalization to establish a rate of convergence in the sense of Marcinkiewicz-Zygmund-type strong law of large numbers.
Theorem . Letα> /,αp> and{Xn,n≥}be a sequence of i.i.d.random variables.
Assume that EX= ifα≤.Then the following statements are equivalent
(i) E|X|p<∞;
(ii) ∞n=nαp–P(max ≤k≤n|
k
i=Xi|>εn
α) <∞for allε> .
Many authors have extended Theorem . for the i.i.d. case to some dependent cases. For example, Shao [] investigated the moment inequalities for theϕ-mixing random variables and gave its application to the complete convergence for this stochastic process; Yu [] ob-tained the complete convergence for weighted sums of martingale differences; Ghosal and Chandra [] gave the complete convergence of martingale arrays; Stoica [, ] investigated the Baum-Katz-Nagaev-type results for martingale differences and the rate of convergence
in the strong law of large numbers for martingale differences; Wanget al.[] also studied the complete convergence and complete moment convergence for martingale differences, which generalized some results of Stoica [, ]; Yanget al.[] obtained the complete con-vergence for the moving average process of martingale differences and so forth. For other works about convergence analysis, one can refer to Gut [], Chenet al.[], Sung [–], Sung and Volodin [], Huet al.[] and the references therein.
Recently, Thanh and Yin [] studied the complete convergence for randomly weighted sums of independent random elements in Banach spaces. On the other hand, Cabreraet al.[] investigated some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables. Inspired by the papers above, we will investigate the complete moment convergence for randomly weighted sums of martingale differences in this paper, which implies the com-plete convergence and Marcinkiewicz-Zygmund-type strong law of large numbers for this stochastic process. We generalize the results of Stoica [, ] and Wanget al.[] for the nonweighted sums of martingale differences to the case of randomly weighted sums of martingale differences. For the details, one can refer to the main results presented in Sec-tion . The proofs of the main results are presented in SecSec-tion .
Recall that the sequence{Xn,n≥}is stochastically dominated by a nonnegative
ran-dom variableXif
sup
n≥
P|Xn|>t≤CP(X>t) for some positive constantCand for allt≥.
Throughout the paper, letF={∅,},x+=xI(x≥),I(B) be the indicator function of
setBandC,C,C, . . . denote some positive constants not depending onn, which may be
different in various places.
To prove the main results of the paper, we need the following lemmas.
Lemma .(cf.Hall and Heyde [], Theorem .) If{Xi,Fi, ≤i≤n}is a martingale
difference and p> ,then there exists a constant C depending only on p such that
E
max
≤k≤n
k
i=
Xi
p ≤C E
n
i=
EXi|Fi–
p/ +E
max
≤i≤n|Xi|
p, n≥.
Lemma .(cf.Sung [], Lemma .) Let{Xn,n≥}and{Yn,n≥}be sequences of
ran-dom variables.Then for any n≥,q> ,ε> and a> ,
E
max
≤j≤n
j
i=
(Xi+Yi)
–εa
+ ≤
εq +
q–
aq–E
max
≤j≤n
j
i=
Xi
q
+E
max
≤j≤n
j
i=
Yi
.
Lemma .(cf.Wanget al.[], Lemma .) Let{Xn,n≥}be a sequence of random
a> and b> ,the following two statements hold
E|Xn|aI|Xn| ≤b≤C
EXaI(X≤b)+baP(X>b)
and
E|Xn|aI|Xn|>b≤CE
XaI(X>b),
where Cand Care positive constants.
2 Main results
Theorem . Let α > /, < p< , ≤ αp< and {Xn,Fn,n ≥ } be a
martin-gale difference sequence stochastically dominated by a nonnegative random variable X with EXp<∞.Assume that{A
n,n≥} is a random sequence,and it is independent of {Xn,n≥}.If
n
i=
EAi =O(n), (.)
then for everyε> ,
∞
n=
nαp––αE
max
≤k≤n
k
i=
AiXi
–εnα
+
<∞ (.)
and forαp> ,
∞
n=
nαp–E
sup
k≥n
ki=AiXi
kα
–ε
+
<∞. (.)
Theorem . Letα> /,p≥and{Xn,Fn,n≥}be a martingale difference sequence
stochastically dominated by a nonnegative random variable X with EXp<∞.Let{An,n≥
}be a random sequence,which is independent of{Xn,n≥}.DenoteG={∅,}andGn= σ(X, . . . ,Xn),n≥.For some q>(ααp–)– ,we assume that E[supn≥E(Xn|Gn–)]q/<∞and
n
i=
E|Ai|q=O(n). (.)
Then for everyε> , (.)and(.)hold.
Meanwhile, for the casep= , we have the following theorem.
Theorem . Letα> and{Xn,Fn,n≥}be a martingale difference sequence
stochas-tically dominated by a nonnegative random variable X with E[Xln( +X)] <∞.Assume that(.)holds and{An,n≥}is a random sequence,which is independent of{Xn,n≥}.
Then for everyε> ,
∞
n=
n–E
max
≤k≤n
k
i=
AiXi
–εnα
+
and forα> ,
∞
n=
nα–E
sup
k≥n
ki=AiXi
kα
–ε
+
<∞. (.)
In particular,forα> ,it has
∞
n=
nα–P
max
≤k≤n
k
i=
AiXi
>εnα
<∞, (.)
and forα> ,it has
∞
n=
nα–P
sup
k≥n
ki=AiXi
kα
>ε
<∞. (.)
On the other hand, forα≥ andEX<∞, we have the following theorem.
Theorem . Letα≥and{Xn,Fn,n≥}be a martingale difference sequence
stochasti-cally dominated by a nonnegative random variable X with EX<∞.DenoteG={∅,}and
Gn=σ(X, . . . ,Xn),n≥.Let(.)hold,and let{An,n≥}be a random sequence,which is
independent of{Xn,n≥}.We assume(i)under the case ofα= ,there exists aδ> such
that
lim
n→∞
max≤i≤nE[|Xi|+δ|Gi–]
nδ = , a.s.
and(ii)under the case ofα> ,it has for anyλ> that
lim
n→∞
max≤i≤nE[|Xi||Gi–]
nλ = , a.s.
Then forα≥and everyε> ,it has(.).In addition,forα> ,it has(.).
Remark . If the conditions of Theorem . or Theorem . hold, then for everyε> ,
∞
n=
nαp–P
max
≤k≤n
k
i=
AiXi
>εnα
<∞, (.)
and forαp> ,
∞
n=
nαp–P
sup
k≥n
ki=AiXi
kα
>ε
<∞. (.)
In fact, it can be checked that for everyε> ,
∞
n=
nαp––αE
max
≤k≤n
k
i=
AiXi
–εnα
+
=
∞
n=
nαp––α
∞
P
max
≤k≤n
k
i=
AiXi
–εnα>t
≥
∞
n=
nαp––α εnα
P
max
≤k≤n
k
i=
AiXi
–εnα>t
dt
≥ε
∞
n=
nαp–P
max
≤k≤n
k
i=
AiXi
> εnα
. (.)
So (.) implies (.).
On the other hand, by the proof of Theorem . of Gut [] and the proof of (.) in Yanget al.[], forαp> , it is easy to see that
∞
n=
nαp–P
sup
k≥n
ki=AiXi
kα
> αε
≤C
∞
n=
nαp–P
max
≤k≤n
k
i=
AiXi
>εnα
.
Thus (.) follows from (.).
Remark . In Theorem ., ifα= /p, then for everyε> , we get by (.) that
∞
n=
n–P
max
≤k≤n
k
i=
AiXi
>εn/p
<∞. (.)
By using (.), one can easily get the Marcinkiewicz-Zygmund-type strong law of large numbers of randomly weighted sums of martingale difference as following
lim
n→∞
n/p
n
i=
AiXi= , a.s.
IfAn=anis non-random (the case of constant weighted),n≥, then one can get the
results of Theorems .-. for the non-random weighted sums of martingale differences. Meanwhile, it can be seen that our condition E[supn≥E(X
n|Gn–)]q/ <∞ in
Theo-rem . is weaker than the conditionsupn≥E(Xn|Fn–)≤C, a.s. in Theorem .,
The-orem . and TheThe-orem . of Wanget al.[]. In fact, it follows fromGn–⊆Fn–that
EXn|Gn–
=EEXn|Fn–
|Gn–
≤E
sup
n≥
EXn|Fn–
|Gn–
.
If supn≥E(Xn|F
n–)≤C, a.s., then it has E[supn≥E(Xn| Gn–)]q/<∞. For α ≥ and
E[Xln( +X)] <∞, Wanget al.[] obtained the result of (.) (see Theorem . of Wang et al.[]). Therefore, by Theorems .-. in this paper, we generalize Theorems .-. of Wanget al.[] for the nonweighted sums of martingale differences to the case of randomly weighted sums of martingale differences.
On the other hand, let the hypothesis that{An,n≥}is independent of{Xn,n≥}be
replaced by that AnisFn–-measurable andAn is independent ofXnfor eachn≥ in
Theorem ., and the other conditions of Theorem . hold, one can get (.) and (.) (the proof is similar to the one of Theorem .). LetAnbeFn–-measurable,Anbe independent
ofXnfor eachn≥,E[supn≥E(Xn|Fn–)]q/<∞and other conditions of Theorem .
hold, one can also obtain (.) and (.). We can obtain some similar results if we only requireAnisFn–-measurable for alln≥ (without any independence hypothesis). This
3 The proofs of main results
Proof of Theorem. LetG={∅,}, forn≥,Gn=σ(X, . . . ,Xn) and
Xni=XiI
|Xi| ≤nα, ≤i≤n.
It can be seen that
AiXi=AiXiI
|Xi|>nα+A
iXni–E(AiXni|Gi–)
+E(AiXni|Gi–), ≤i≤n.
So, by Lemma . witha=nα, forq> , one has that
∞
n=
nαp––αE
max
≤k≤n
k
i=
AiXi
–εnα
+
≤C
∞
n=
nαp––qαE
max
≤k≤n
k
i=
AiXni–E(AiXni|Gi–)
q
+
∞
n=
nαp––αE
max
≤k≤n
k
i=
AiXiI
|Xi|>nα+E(AiXni|Gi–)
≤C
∞
n=
nαp––qαE
max
≤k≤n
k
i=
AiXni–E(AiXni|Gi–)
q
+
∞
n=
nαp––αE
max
≤k≤n
k
i=
AiXiI
|Xi|>nα
+
∞
n=
nαp––αE
max
≤k≤n
k
i=
E(AiXni|Gi–)
:=H+H+H. (.)
Obviously, it follows from Hölder’s inequality and (.) that
n
i=
E|Ai| ≤
n
i=
EAi
/n
i=
/
=O(n). (.)
By the fact that{An,n≥}is independent of{Xn,n≥}, we can check by Markov’s
in-equality, Lemma ., (.) andEXp<∞(p> ) that
H≤
∞
n=
nαp––α
n
i=
E|Ai|E|Xi|I|Xi|>nα
≤
∞
n=
nαp––αEXIX>nα
=
∞
n=
nαp––α
∞
m=n
=
∞
m=
EXImα<X≤(m+ )α
m
n=
nαp––α
≤C
∞
m=
mαp–αEXImα<X≤(m+ )α
≤CEXp<∞. (.)
On the other hand, one can see that{Xn,Gn,n≥}is also a martingale difference, since {Xn,Fn,n≥}is a martingale difference. Combining with the fact that{An,n≥}is
inde-pendent of{Xn,n≥}, we have that
E(AnXn|Gn–) =E
E(AnXn|Gn)|Gn–
=EXnE(An|Gn)|Gn–
=EAnE[Xn|Gn–] = , a.s.,n≥.
Consequently, by the proof of (.), it follows that
H =
∞
n=
nαp––αE
max
≤k≤n
k
i=
EAiXiI
|Xi| ≤nα|Gi–
=
∞
n=
nαp––αE
max
≤k≤n
k
i=
EAiXiI
|Xi|>nα|Gi–
≤
∞
n=
nαp––α
n
i=
E|Ai|E|Xi|I|Xi|>nα
≤C
∞
n=
nαp––αEXIX>nα≤CEXp<∞. (.)
Next, we turn to proveH<∞. It can be found that for fixed real numbersa, . . . ,an,
aiXni–E(aiXni|Gi–),Gi, ≤i≤n
is also a martingale difference. Note that {A,A, . . . ,An} is independent of {Xn,Xn,
. . . ,Xnn}. So, by Markov’s inequality, (.), (.) with q= , Lemma . withp= and Lemma ., we get that
H =C
∞
n=
nαp––αE E max
≤k≤n k
i=
aiXni–E(aiXni|Gi–)
A=a, . . . ,An=an
≤CE
n
i=
E(aiXni)
A=a, . . . ,An=an
=C
∞
n=
nαp––α
n
i=
E(AiXni)=C
∞
n=
nαp––α
n
i=
EAiEXni
≤C
∞
n=
nαp––αEXIX≤nα+C
∞
n=
nαp–PX>nα
By the conditionEXp<∞withp< , it follows
H=
∞
n=
nαp––α
n
i=
EXI(i– )α<X≤iα
=
∞
i=
EXI(i– )α<X≤iα
∞
n=i
nαp––α
≤C
∞
i=
EXpX–pI(i– )α<X≤iαiαp–α≤C
EXp<∞. (.)
From (.), it has
H≤
∞
n=
nαp––αEXIX>nα≤CEXp<∞. (.)
Consequently, by (.) and (.)-(.), we obtain (.) immediately. Forαp> , we turn to prove (.). DenoteSk =
k
i=AiXi, k≥. It can be seen that αp< < +α. So, similar to the proof of (.) in Yanget al.[], we can check that
∞
n=
nαp–E
sup
k≥n
Sk
kα
–εα
+ ≤C
∞
l=
l(αp––α)
∞
P
max
≤k≤l|Sk|>ε
α(l+)+sds
≤C+α–αp
∞
n=
nαp––αE
max
≤k≤n|Sk|–εn
α+
.
Combining with (.), we get (.) finally.
Proof of Theorem. To prove Theorem ., we use the same notation as that in the proof of Theorem .. Forp≥, it is easy to see thatq> (αp– )/(α– )≥. Consequently, for any ≤s≤, by Hölder’s inequality and (.), we get
n
i=
E|Ai|s≤
n
i=
E|Ai|q
s/qn
i=
–s/q
=O(n). (.)
By (.), (.) and (.), one can find thatH<∞andH<∞. So we need to prove that
H<∞under the conditions of Theorem .. Forp≥, noting that{A,A, . . . ,An}is
independent of{Xn,Xn, . . . ,Xnn}, similar to the proof of (.), one has by Lemma . that
H =C
∞
n=
nαp––qαE
max
≤k≤n
k
i=
AiXni–E(AiXni|Gi–)
q
≤C
∞
n=
nαp––qαE
n
i=
EAiXni–E(AiXni|Gi–)
|Gi–
q/
+C
∞
n=
nαp––qα
n
i=
EAiXni–E(AiXni|Gi–) q
Obviously, for ≤i≤n, it has
EAiXni–E(AiXni|Gi–)
|Gi–
=EAiXiI|Xi| ≤nα|G
i–
–EAiXiI
|Xi| ≤nα|G
i–
≤EAiXiI|Xi| ≤nα|Gi–
≤EAiEXi|Gi–
, a.s.
Combining (.) withE[supi≥E(X
i|Gi–)]q/<∞, we obtain that
H≤
∞
n=
nαp––qα
n
i=
EAi q/
E
sup
i≥
EXi|Gi–
q/
≤C
∞
n=
nαp––qα+q/<∞, (.)
following from the fact thatq> (αp– )/(α– ). Meanwhile, byCrinequality, Lemma .
and (.),
H≤C
∞
n=
nαp––qα
n
i=
E|Ai|qE|Xi|qI|Xi| ≤nα
≤C
∞
n=
nαp––qαEXqIX≤nα+C
∞
n=
nαp–PX>nα
≤C
∞
n=
nαp––qαEXqIX≤nα+C
∞
n=
nαp––αEXIX>nα
=:CH∗ +CH∗. (.)
By the conditionp≥ andα> /, we have that (αp– )/(α– ) –p≥, which implies thatq>p. So, one gets byEXp<∞that
H∗ =
∞
n=
nαp––qα
n
i=
EXqI(i– )α<X≤iα
=
∞
i=
EXqI(i– )α<X≤iα
∞
n=i
nαp––qα
≤C
∞
i=
EXpXq–pI(i– )α<X≤iαiαp–qα≤CEXp<∞. (.)
By the proof of (.), it follows
H∗ =
∞
n=
nαp––αEXIX>nα≤CEXp<∞. (.)
Therefore, by (.)-(.), it hasH<∞. Consequently, it completes the proof of (.).
Proof of Theorem. Similar to the proof of Theorem ., by Lemma ., it can be checked that
∞
n=
n–E
max
≤k≤n
k
i=
AiXi
–εnα
+
≤C
∞
n=
n––αE
max
≤k≤n
k
i=
AiXni–E(AiXni|Gi–)
+
∞
n=
n–E
max
≤k≤n
k
i=
AiXiI
|Xi|>nα
+
∞
n=
n–E
max
≤k≤n
k
i=
E(AiXni|Gi–)
:=J+J+J. (.)
Similarly to the proof of (.), we have
J≤C
∞
n=
n–EXIX>nα
=C
∞
n=
n–
∞
m=n
EXImα<X≤(m+ )α
=C
∞
m=
EXImα<X≤(m+ )α
m
n=
n–
≤C
∞
m=
ln( +m)EXImα<X≤(m+ )α
≤CE
Xln( +X)<∞. (.)
Meanwhile, by the proofs of (.) and (.), we get
J≤C
∞
n=
n–EXIX>nα≤C
E
Xln( +X)<∞. (.)
On the other hand, by the proof of (.), it can be checked that forα> ,
J≤C
∞
n=
n––α
n
i=
E(AiXni)=C
∞
n=
n––α
n
i=
EAiEXni
≤C
∞
n=
n––αEXIX≤nα+C
∞
n=
nα–PX>nα
≤C
∞
n=
n––α
n
i=
EXI(i– )α<X≤iα+C
∞
n=
n–EXIX>nα
≤C
∞
i=
EXI(i– )α<X≤iα
∞
n=i
n––α+CE
≤C
∞
i=
EXI(i– )α<X≤iαi–α+CE
Xln( +X)
≤CEX+CE
Xln( +X)<∞. (.)
Therefore, by (.)-(.), one gets (.) immediately. Similar to the proof of (.), it is easy to have (.). Obviously, by the proof of (.) in Remark ., (.) also holds under the conditions of Theorem .. Finally, by the proof of Theorem . of Gut [] and the proof of (.) in Yanget al.[], forα> , it is easy to get (.).
Proof of Theorem. Forn≥, we also denoteXni=XiI(|Xi| ≤nα), ≤i≤n. It is easy to
see that
P
max
≤k≤n
k i=
AiXi
>εnα
≤ n
i=
P|Xi|>nα+P
max
≤k≤n
k i=
AiXni
>εnα
. (.)
For the case ofα= , there exists aδ> such thatlimn→∞max≤i≤nE[|Xi|+δ|Gi–]
nδ = , a.s. So byE(AnXn|Gn–) = , a.s.,n≥, we can check that
nα
max
≤k≤n
k i=
E(AiXni|Gi–)
= n max
≤k≤n
k i=
EAiXiI
|Xi| ≤n|Gi–
= n max
≤k≤n
k i=
EAiXiI
|Xi|>n|Gi–
≤ n n i=
E|Ai|E|Xi|I|Xi|>n|Gi–
≤
n+δ
n
i=
E|Ai|E|Xi|+δ|Gi–
≤ K
nδmax≤i≤nE
|Xi|+δ|Gi–
→, a.s.,
asn→ ∞.
Otherwise, for the case ofα> , it is assumed thatlimn→∞max≤i≤nE[|Xi||Gi–]
nλ = , a.s., for anyλ> . Consequently, for anyα> , it follows that
nα
max
≤k≤n
k i=
E(AiXni|Gi–)
= nα max
≤k≤n
k i=
EAiXiI
|Xi| ≤n|Gi–
≤ nα n i=
E|Ai|E|Xi|I|Xi| ≤n|Gi–
≤ K
nα–max≤i≤nE[|Xi||Gi–]→, a.s.,
asn→ ∞. Meanwhile,
∞
n=
nα–
n
i=
P|Xi|>nα≤K
∞
n=
By (.) and (.), to prove (.), it suffices to show that
I=
∞
n=
nα–P
max
≤k≤n
k
i=
AiXni–E(AiXni|Gi–)
>
εnα
<∞.
Obviously, by Markov’s inequality and the proofs of (.), (.), (.), one can check that
I≤
ε
∞
n=
n––αE
max
≤k≤n
k
i=
AiXni–E(AiXni|Gi–)
≤K
∞
n=
n––αEXIX≤nα+K
∞
n=
nα–PX>nα
≤KEX<∞.
On the other hand, by proof of Theorem . of Gut [] and the proof of (.) in Yang
et al.[], we can easily obtain (.) forα> .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1School of Mathematical Science, Anhui University, Hefei, China.2Department of System Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong, China.
Acknowledgements
The authors are most grateful to editor prof. Soo Hak Sung and two anonymous referees for their careful reading and insightful comments, which helped to significantly improve an earlier version of this paper. Supported by the NNSF of China (11171001, 11201001), Natural Science Foundation of Anhui Province (1208085QA03, 1308085QA03), Talents Youth Fund of Anhui Province Universities (2012SQRL204) and Doctoral Research Start-up Funds Projects of Anhui University.
Received: 31 March 2013 Accepted: 6 August 2013 Published: 21 August 2013 References
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doi:10.1186/1029-242X-2013-396
