R E S E A R C H
Open Access
On strong law of large numbers and growth
rate for a class of random variables
Yan Shen, Jie Yang and Shuhe Hu
**Correspondence:
School of Mathematical Science, Anhui University, Hefei, 230039, P.R. China
Abstract
In this paper, we study the strong law of large numbers for a class of random variables satisfying the maximal moment inequality with exponent 2. Our results embrace the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for this class of random variables. In addition, strong growth rate for weighted sums of this class of random variables is presented.
MSC: 60F15
Keywords: strong law of large numbers; weighted sums; with exponent 2
1 Introduction
Let{Xn,n≥}be a sequence of random variables defined on a fixed probability space (,F,P).Sn=
n
i=Xi,n≥,S= . Let{an,n≥}and{bn,n≥}be sequences of con-stant with <bn↑ ∞. Then{anXn,n≥}is said to obey the general strong law of large numbers (SLLN) with norming constant{bn,n≥}if the normed weighted sums
bn n
i=
ai(Xi–EXi)→ almost surely (a.s.) (.)
holds. Note that the SLLN of the form (.) embraces the Kolmogorov SLLN (bn=n,an= ) and the Marcinkiewicz SLLN (bn=n/r,an= ,r> ). Whenbn=
n
i=ai, fundamental results for the SLLN were obtained.
Under an independent assumption, many SLLNs for the weighted sums are obtained. One can refer to Adler and Rosalsky [], Chow and Teicher [], Fernholz and Teicher [], Jamisonet al.[] and Teicher [].
Under a pairwise independent assumption, Rosalsky [] obtained some SLLNs for weighted sums of pairwise independent and identically distributed random variables. Sung [] obtained sufficient conditions for (.) if{Xn,n≥} is a sequence of pairwise independent random variables satisfying∞xp–supn≥P(|Xn|>x)dx<∞. Sung [] pre-sented the following result:a
n
n
i=(Xi–EXiI(|Xi| ≤ai))→ a.s., where{an}is a sequence of positive constants withan
n ↑and{Xn}is a sequence of pairwise independent and iden-tically distributed random variables.
For more details about strong limit theorems for dependent case, one can refer to Wu [], Wu and Jiang [], Huet al.[], Shenet al.[], Zhouet al.[] and Zhou [], and so forth.
Recently Sung [] gave the following definition.
Definition .(Sung []) A random variable sequence{Xn,n≥}is said to satisfy the maximal moment inequality with exponent if for alln≥m≥, there exists a constant
Cindependent ofnandmsuch that
E
max
m≤k≤n
k
i=m
Xi
≤C
n
i=m
EXi. (.)
We can see that a wide class of mean zero random variables satisfies (.). Inspired by Sung [, ], we establish SLLN of the form (.) for a class of random variables satisfying the maximal moment inequality with exponent .
The rest of the paper is organized as follows. In Section , some preliminary definition and lemmas are presented. In Section , main results and their proofs are provided.
Throughout the paper, letI(A) be the indicator function of the setA.Cdenotes a positive constant not depending on n, which may be different in various places. Let{an,n≥}
and{bn,n≥}be sequences of positive numbers,anbnrepresents that there exists a constantC> such thatan≤Cbnfor alln.
2 Preliminaries
The following lemmas and definition will be needed in this paper.
Lemma .(Sung []) Let{Xn,n≥}be a sequence of random variables and put G(x) =
supn≥P(|Xn|>x)for x≥.Assume that∞xp–G(x)dx<∞for some≤p< .Then
(i) ∞n=P(|Xn|>n/p) <∞.
(ii) ∞n=EX
nI(|Xn| ≤n/p)/n/p<∞.
(iii) EXnI(|Xn|>cn)→for any sequence{cn,n≥}satisfyingcn→ ∞.
Lemma .(Sung []) Let{Xn,n≥}be a sequence of random variables satisfying the
maximal moment inequality with exponent.If∞n=EXn<∞,then∞n=Xnconverges
almost surely.
Definition . A random variable sequence{Xn,n≥}is said to be stochastically
domi-nated by a random variableXif there exists a constantCsuch that
P|Xn|>x ≤CP|X|>x
for allx≥ andn≥.
Lemma . Let{Xn,n≥}be a sequence of random variables which is stochastically
dom-inated by a random variable X.For anyα> and b> ,the following statement holds:
E|Xn|αI|Xn| ≤b ≤CE|X|αI|X| ≤b +bαP|X|>b , (.)
E|Xn|αI|Xn|>b ≤CE|X|αI|X|>b , (.)
Lemma .(Hu []) Let b,b, . . . be a nondecreasing unbounded sequence of positive
numbers.Letα,α, . . .be nonnegative numbers,andk=α+· · ·+αkfor k≥.Let r be a
fixed positive number.Assume that for each n≥,
E
max
≤k≤n|Sk|
r ≤C
n
k=
αk. (.)
If
∞
l=
l
br l
–
br l+
<∞, (.)
n
br n
is bounded, (.)
then
lim
n→∞
Sn
bn
= a.s., (.)
and with the growth rate
Sn
bn =O
βn
bn
a.s., (.)
where
βn=max ≤k≤nbkν
δ/r
k , ∀ <δ< ,vn= ∞
k=n
αk
br k
, lim
n→∞
βn
bn
= . (.)
And
E
max
≤l≤n
Sl
bl
r≤C
n
l=
αl
brl <∞, (.)
E
sup
l≥
Sl
bl
r≤C
∞
l=
αl
br l
<∞. (.)
If we further assume thatαn> for infinitely many n,then
E
sup
l≥
Sl
βl
r≤C
∞
l=
αl
βr l
<∞. (.)
Proof It follows from Corollary .. of Hu [] that (.)-(.) hold. By (.) and Theo-rem . of Fazekas and Klesov [], we have
E
max
≤l≤n
Sl
bl
r≤C
n
l=
αl
br l
≤C
∞
l=
αl
br l
Therefore
E
sup
l≥
Sl
bl
r= lim
n→∞E
max
≤l≤n
Sl
bl
r≤C
∞
l=
αl
br l
<∞, (.)
following from the monotone convergence theorem of Rao []. Equation (.) follows
from the proof of Lemma . of Hu and Hu [].
3 Main results
Theorem . Let {Xn,n ≥ } be a sequence of random variables and put G(x) =
supn≥P(|Xn|>x)for x> .Denote Yn= –n/pI(Xn< –n/p) +XnI(|Xn| ≤n/p) +n/pI(Xn>
np),n≥,where p is a positive constant.Let{an,n≥}and{bn,n≥}be sequences of
positive numbers with bn↑ ∞.Suppose that{abnn(Yn–EYn),n≥}satisfies the maximal
moment inequality with exponent.Assume that the following two conditions hold:
n
i=
ai=O(bn), (.)
an
bn
=On–/p for some≤p< . (.)
If∞xp–G(x)dx<∞,then
lim
n→∞
bn n
i=
ai(Xi–EXi) = a.s. (.)
Proof By Lemma .(i),
∞
n=
P(Xn =Yn) = ∞
n=
P|Xn|>n/p <∞. (.)
ThereforeP(Xn=Yn, i.o.) = follows from the Borel-Cantelli lemma and (.). Thus (.) is equivalent to the following:
lim
n→∞
bn n
i=
ai(Yi–EXi) = a.s. (.)
So, in order to prove (.), we need only to prove
lim
n→∞
bn n
i=
ai(Yi–EYi) = a.s. (.)
and
lim
n→∞
bn n
i=
Firstly, we prove (.). In view of Lemma .(i), (ii) and (.), we have
∞
n=
E
an
bn
(Yn–EYn)
≤
∞
n=
a n
b n
EYn
= ∞
n=
a n
b n
EXnI|Xn| ≤n/p +n/pEI|Xn|>n/p
∞
n=
n–/pEXnI|Xn| ≤n/p +P|Xn|>n/p
<∞.
Thus it follows by Lemma . that
∞
n=
an
bn
(Yn–EYn) converges a.s. (.)
By Kronecker’s lemma, we can obtain (.) immediately.
Secondly, we prove (.). By (.) and Lemma .(iii), we can get
lim
n→∞
bn n
i=
aiEXiI
|Xi|>i/p = . (.)
By (.) and Lemma .(i), we have
∞
n=
an
bn
n/pPXn>n/p ∞
n=
PXn>n/p < ∞
n=
P|Xn|>n/p <∞
and
∞
n=
an
bn
n/pPXn< –n/p <∞.
Thus it follows by Kronecker’s lemma that
lim
n→∞
bn n
i=
aii/pP
Xi>i/p = (.)
and
lim
n→∞
bn n
i=
aii/pP
Xi< –i/p = . (.)
Therefore,
lim
n→∞
bn n
i=
ai(EXi–EYi)
= lim
n→∞
bn n
i=
aiEXiI
+
bn n
i=
aii/pP
Xi< –i/p –
bn n
i=
aii/pP
Xi>i/p
=
follows from (.)-(.). Hence the result is proved.
Theorem . Let {Xn,n≥} be a sequence of mean zero random variables, which is
stochastically dominated by a random variable X.Let{an,n≥} and{bn,n≥} be se-quences of positive numbers with bn↑ ∞.Put cn= abnn for n≥, <r< .Denote Yn= –cnI(Xn< –cn) +XnI(|Xn| ≤cn) +cnI(Xn>cn),n≥,and suppose that{bann(Yn–EYn),n≥}
satisfies the maximal moment inequality with exponent.Assume that the following two conditions hold:
E|X|r<∞, (.)
N(n) =Card{i:ci≤n} nr, n≥. (.)
Then
lim
n→∞
bn n
i=
aiXi= a.s. (.)
Proof LetN() = . By (.), we can see thatcn→ ∞asn→ ∞. By (.) and (.),
∞
n=
P(Xn =Yn) = ∞
n=
P|Xn|>cn ∞
n=
P|X|>cn
= ∞
n=
cn≤j<cn+
P|X|>cn
≤ ∞
j=
j–<cn≤j
P|X|>j–
= ∞
j=
N(j) –N(j– ) ∞
k=j
Pk– <|X| ≤k
= ∞
k=
Pk– <|X| ≤k
k
j=
N(j) –N(j– )
= ∞
k=
N(k)Pk– <|X| ≤k
∞
k=
krPk– <|X| ≤k
which impliesP(Xi =Yi, i.o.) = from the Borel-Cantelli lemma. So, in order to prove (.), we need only to prove
lim
n→∞
bn n
i=
aiYi= a.s. (.)
By (.), (.), Lemma ., and the proof of (.), we have
∞
n=
E
an
bn
(Yn–EYn)
≤ ∞
n=
c–n EYn
= ∞
n=
c–n EXnI|Xn| ≤cn +E
cnI|Xn|>cn
∞
n=
c–n EXI|X| ≤cn + ∞
n=
P|X|>cn
= ∞
n=
cn≤j<cn+
c–n EXI|X| ≤cn + ∞
n=
P|X|>cn
≤ ∞
j=
j–<cn≤j
c–n EXI|X| ≤j +C
∞
j=
N(j) –N(j– )(j– )– j
k=
EXIk– <|X| ≤k +C
= ∞
j=
N(j) –N(j– )(j– )–
EXI <|X| ≤ + j
k=
EXIk– <|X| ≤k
+C
= ∞
j=
N(j) –N(j– )(j– )–EXI <|X| ≤
+ ∞
k=
EXIk– <|X| ≤k
∞
j=k
N(j) –N(j– ) (j– )–+C
≤ ∞
j=
N(j)(j– )––j– EXI <|X| ≤
+ ∞
k=
EXIk– <|X| ≤k
∞
j=k
N(j)(j– )––j– +C
∞
j=
jr–+ ∞
k=
EXIk– <|X| ≤k
∞
j=k
jr–+C
∞
k=
kr–E|X|rk–rIk– <|X| ≤k +C
Combining Lemma ., (.) and Kronecker’s lemma, we can get
lim
n→∞
bn n
i=
ai(Yi–EYi) = a.s. (.)
To complete the proof of (.), it suffices to show that
lim
n→∞
bn n
i=
aiEYi= . (.)
By (.), (.) andEXn= , it follows that
∞
n=
an
bn
EYn
≤
∞
n=
c–n E|Xn|I|Xn|>cn +E
cnI
|Xn|>cn
= ∞
n=
c–n E|Xn|I|Xn|>cn + ∞
n=
P|Xn|>cn . (.)
Observe that
∞
n=
c–nE|Xn|I|Xn|>cn
≤C
∞
n=
c–n E|X|I|X|>cn
=C
∞
n=
cn≤j<cn+
c–nE|X|I|X|>cn
≤C
∞
j=
j–<cn≤j
c–n E|X|I|X|>j–
≤C
∞
j=
N(j) –N(j– )(j– )– ∞
n=j–
E|X|In<|X| ≤n+
=C
∞
n=
E|X|In<|X| ≤n+ n+
j=
N(j) –N(j– )(j– )–
≤C
∞
n=
E|X|In<|X| ≤n+ n
j=
N(j)(j– )––j–
+C
∞
n=
E|X|In<|X| ≤n+ N(n+ )
n
≤C
∞
n=
E|X|In<|X| ≤n+ n
j=
jr–+C
∞
n=
nr–E|X|In<|X| ≤n+
≤C
∞
n=
≤C
∞
n=
E|X|rIn<|X| ≤n+
≤CE|X|r<∞. (.)
So, we can get
∞
n=
an
bn
EYn
<∞
from (.), (.) and (.). Consequently,
∞
n=
an
bn
EYn converges, (.)
which implies (.) from Kronecker’s lemma. We complete the proof of theorem.
Theorem . Let{Xn,n≥} be a sequence of mean zero random variables satisfying
the maximal moment inequality with exponent.Denote Qn=max≤k≤nEXk,n≥and
Q= .For≤p< ,assume that
∞
n=
Qn
n/p <∞. (.)
Then
lim
n→∞
Sn
n/p = a.s., (.)
and with the growth rate
Sn
n/p =O
βn
n/p
a.s., (.)
where
βn=max ≤k≤nk
/pνδ/
k , ∀ <δ< ,vn= ∞
k=n
αk
k/p,
αk=C
kQk– (k– )Qk– , k≥,n→∞lim
βn
n/p = .
(.)
And
E
max
≤l≤n
Sl
l/p
≤
n
l=
αl
l/p <∞, (.)
E
sup
l≥
Sl
l/p
≤
∞
l=
αl
If we further assume thatαn> for infinitely many n,then E sup l≥ Sl βl ≤ ∞ l= αl
βl <∞. (.)
In addition,for any <r< ,
E sup l≥ Sl
l/p
r≤ + r –r
∞
l=
αl
l/p <∞. (.)
Proof Since{Xn,n≥}is a sequence of mean zero random variables satisfying the maxi-mal moment inequality with exponent , we have
E max ≤j≤n j k= Xk ≤C n k=
EXk≤CnQn= n
k=
αk. (.)
And we can obtainαk≥ for allk≥ from its definition. Denotebn=n/p andn=
n
k=αk,n≥. By (.), we can get ∞ l= l b l – b l+ =C ∞ l= lQl
l/p – (l+ )/p
≤C
p
∞
l=
Ql
l/p <∞. (.)
Thus (.) holds. It follows from Remark . in [] that (.) implies (.). By Lemma ., we can get (.)-(.) immediately. It follows from (.) that
E sup l≥ Sl
l/p
r=
∞ P sup l≥ Sl
l/p
r>t
dt = P sup l≥ Sl
l/p
r>t
dt+ ∞ P sup l≥ Sl
l/p
r>t
dt
≤ +E
sup
l≥
Sl
l/p
∞t–/rdt
≤ + r –r
∞
l=
αl
l/p <∞.
The proof is completed.
Remark . It is easy to see that a wide class of mean zero random variables satisfies the maximal moment inequality with exponent . Examples include independent random variables, negatively associated random variables (see Matula []), negatively superad-ditive dependent random variables (see Shenet al.[]),ϕ-mixing random variables and AANA random variables (see Wanget al.[, ]), andρ˜-mixing random variables (see Utevet al.[]). So Theorems .-. hold for this wide class of random variables.
Competing interests
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors are most grateful to the editor and the anonymous referee for their careful reading and insightful comments. This work is supported by the National Natural Science Foundation of China (11171001, 11201001), Natural Science Foundation of Anhui Province (1208085QA03), Humanities and Social Sciences Project from Ministry of Education of China (12YJC91007), Key Program of Research and Development Foundation of Hefei University (13KY05ZD) and Doctoral Research Start-up Funds Projects of Anhui University.
Received: 9 August 2013 Accepted: 4 November 2013 Published:25 Nov 2013
References
1. Adler, A, Rosalsky, A: On the strong law of large numbers for normed weighted sums of i.i.d. random variables. Stoch. Anal. Appl.5, 467-483 (1987)
2. Chow, YS, Teicher, H: Almost certain summability of independent, identically distributed random variables. Ann. Math. Stat.42, 401-404 (1971)
3. Fernholz, LT, Teicher, H: Stability of random variables and iterated logarithm laws of martingales and quadratic forms. Ann. Probab.8, 765-774 (1980)
4. Jamison, B, Orey, S, Pruitt, W: Convergence of weighted averages of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.4, 40-44 (1965)
5. Teicher, H: Almost certain convergence in double arrays. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 331-345 (1985) 6. Rosalsky, A: Strong stability of normed weighted sums of pairwise i.i.d. random variables. Bull. Inst. Math. Acad. Sin.
15, 203-219 (1987)
7. Sung, SH: Strong law of large numbers for weighted sums of pairwise independent random variables. Bull. Inst. Math. Acad. Sin.27(1), 23-28 (1999)
8. Sung, SH: On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions. Stat. Probab. Lett.83, 1963-1968 (2013)
9. Wu, QY: A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J. Inequal. Appl.2010, Article ID 383805 (2010)
10. Wu, QY, Jiang, YY: Some strong limit theorems for weighted product sums ofϕ-mixing sequences of random˜ variables. J. Inequal. Appl.2009, Article ID 174768 (2009)
11. Hu, SH, Wang, XJ, Yang, WZ, Zhao, T: The Hàjek-Rènyi-type inequality for associated random variables. Stat. Probab. Lett.79, 884-888 (2009)
12. Shen, Y, Wang, XJ, Yang, WZ, Hu, SH: Almost sure convergence theorem and strong stability for weighted sums of NSD random variables. Acta Math. Sin. Engl. Ser.29, 743-756 (2013)
13. Zhou, XC, Tan, CC, Lin, JG: On the strong laws for weighted sums ofρ*-mixing random variables. J. Inequal. Appl.
2011, Article ID 157816 (2011)
14. Zhou, XC: Complete moment convergence of moving average processes underϕ-mixing assumptions. Stat. Probab. Lett.80, 285-292 (2010)
15. Sung, SH: On the strong law of large numbers for weighted sums of random variables. Comput. Math. Appl.62, 4277-4287 (2011)
16. Hu, SH: Some new results for the strong law of large numbers. Acta Math. Sin. (Chin. Ser.)46(6), 1123-1134 (2003) (Chinese)
17. Fazekas, I, Klesov, O: A general approach to the strong law of large numbers. Theory Probab. Appl.45(3), 436-449 (2002)
18. Rao, MM: Measure Theory and Integration. Wiley, New York (1987)
19. Hu, SH, Hu, M: A general approach rate to the strong law of large numbers. Stat. Probab. Lett.76, 843-851 (2006) 20. Matula, P: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab.
Lett.12, 209-213 (1992)
21. Wang, XJ, Hu, SH, Shen, Y, Yang, WZ: Moment inequality forϕ-mixing sequences and its applications. J. Inequal. Appl. 2009, Article ID 379743 (2011)
22. Wang, XJ, Hu, SH, Yang, WZ: Convergence properties for asymptotically almost negatively associated sequence. Discrete Dyn. Nat. Soc.2010, Article ID 218380 (2010)
23. Utev, S, Peligrad, M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J. Theor. Probab.16, 101-115 (2003)
10.1186/1029-242X-2013-563
Cite this article as:Shen et al.:On strong law of large numbers and growth rate for a class of random variables.