Volume 2011, Article ID 279754,13pages doi:10.1155/2011/279754
Research Article
On Strong Law of Large Numbers for
Dependent Random Variables
Zhongzhi Wang
Faculty of Mathematics and Physics, Anhui University of Technology, Ma’anshan 243002, China
Correspondence should be addressed to Zhongzhi Wang,[email protected]
Received 16 December 2010; Accepted 3 March 2011
Academic Editor: Vijay Gupta
Copyrightq2011 Zhongzhi Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We discuss strong law of large numbers and complete convergence for sums of uniformly bounded negatively associateNArandom variablesRVs. We extend and generalize some recent results. As corollaries, we investigate limit behavior of some other dependent random sequence.
1. Introduction
Throughout this paper, let denote the set of nonnegative integer, let{X, Xn, n ∈ } be a
sequence of random variables defined on probability spaceΩ,F, P, and putSn
n
k1Xk.
The symbolCwill denote a generic constant0 < C <∞which is not necessarily the same one in each appearance.
In1, Jajte studied a large class of summability method as follows: a sequence{Xn, n≥
1}is summable toXby the methodh, gif
1
gn
n
k1
Xk
hk −→X, asn−→ ∞. 1.1
The main result of Jajte is as follows.
Theorem 1.1. Letg·be a positive, increasing function andh·a positive function such thatφy≡
gyhysatisfies the following conditions.
1For somed≥0,φ·is strictly increasing ond,∞with range0,∞.
2There exist C and a positive integerk0such thatφy1/φy≤C,y≥k0.
Then, for i.i.d. random variables{Xn, n∈ },
1
gn
n
k1
Xk−EXk1|Xk|≤φk
hk −→0 a.s.iffE
φ−1|X|<∞, 1.2
whereφ−1is the inverse of functionφ,1Ais the indicator of eventA.
Motivated by Jajte 1, the present paper is devoted to the study of the limiting behavior of sums when{X, Xn, n ∈ } are dependent RVs In particular, we willl consider
the case when{X, Xn, n ∈ } are NA RVs and obtain some general results on the complete
convergence of dependent RVs First, we shall give some definitions.
Definition 1.2. A finite family of random variables{Xn,1 ≤ i ≤ n} is said to be negatively
associatedabbreviated NAif, for every pair of disjoint subsetsAandBof{1,2, . . . , n}, we have
Covf1Xi, i∈A, f2
Xj, j ∈B
≤0, 1.3
wheneverf1andf2are coordinatewise increasing and the covariance exists.
Definition 1.3. A finite family of random variables{Xn,1 ≤ i ≤ n} is said to be positively
associatedabbreviated PAif
CovfX1, . . . , Xn, gX1, . . . , Xn
≥0, 1.4
wheneverfandgare coordinatewise increasing and the covariance exists.
An infinite family of random variables is NAresp., PAif every finite subfamily is NAresp., PA.
LetFn
mbe theσ-algebra generated by RVsXi,m≤i≤n.
Definition 1.4. A sequence of random variables{Xn, n∈ }is said to bem-dependence ifFr1
andF∞r are independent for allrandrsuch that 1≤r < r<∞, r−r > m.
Definition 1.5. A sequence of random variables {Xn, n ∈ } is said to be ϕ-mixing or
uniformly strong mixing, if
ϕτ sup
t∈
sup
A∈Ft
1, B∈F∞tτ, PA>0
|PB|A−PB| −→0 asτ−→ ∞. 1.5
Definition 1.6. Let {Xn, n ∈ } be a sequence of random variables which is said to be:
uniformly bounded by a random variableX we write{Xn, n ∈ } ≺ X if there exists a
constantC >0, for almost everyω∈Ω, such that
sup
n≥1
P{|Xn|> t} ≤CP{|X|> t} ∀t >0. 1.6
Remark 1.7. The uniformly bounded random variables in 1.6can be insured by moment conditions. For example, if
sup
n E|Xn|
2δ<∞ for someδ >0,
1.7
then there exists a uniformly bounded random variableXsuch thatEX2 <∞.
The structure of this paper is as follows. Some needed technical results will be presented in Section 2. The strong law of large numbers for NA RVs will be established in Section3. The Spitzer and Hus-Robbins-type law of large numbers will be presented in Sections4and5, respectively.
2. Preliminaries
We now present some terminologies and lemmas. The following six properties are listed for reference in obtaining the main results in the next sections. Detailed proofs can be founded in the cited references.
Lemma 2.1cf.4 three-series theorem for NA. Let{Xn, n ∈ } be NA. LetC > 0and let
XC
n Xn1|Xn|≤C. In order that
∞
n1Xnconverges a.s., it is sufficient that
1∞n1P{|Xn|> C}<∞,
2∞n1EXnCconverges,
3∞n1varXC n<∞.
Lemma 2.2cf.4. Let{Xn, n∈ }be NA withEXn0,EXn2<∞, then forp >2, for allε >0
P max
1≤k≤n|Sk| ≥ε
≤2ε−2
n
i1
EXi2,
E
max
1≤k≤n|Sk| 2
≤C
n
i1
EX2 i,
E
max
1≤k≤n|Sk| p
≤C
⎧ ⎨ ⎩
n
i1
EXip
n
i1
EXi2
p/2⎫⎬
⎭.
Lemma 2.3cf.5. Let{Xn, n∈ }bem-dependence withEXn0,EX2n<∞, then
ESn2≤m1 n
i1
EXi2. 2.2
Lemma 2.4cf.5. Let{Xn, n∈ }beϕ-mixing withEXn0,EX2n<∞, then
ESn2≤
14
n−1
r1
ϕ1/2r
n
i1
EX2i. 2.3
Lemma 2.5cf.6. Let{Xn, n∈ }be PA withEXn0,EX2n<∞, then
Emax
1≤k≤nS 2
k≤2ES2n. 2.4
Furthermore, if
n
k1
μ1/22n<∞, 2.5
then
Emax
1≤k≤nS 2
k≤Cn1max≤k≤nEXk2, 2.6
whereμn supi≥1j:j−i≥ncovXi, Xj.
Lemma 2.6. Let{Xn, n≥1}be a sequence of random variables andXa random variable. If{Xn} ≺
X, then for allt >0,p≥2
X
p
n1|Xn|≤t≤C
tp{|X|> t}X
p1
|X|≤t. 2.7
Proof. By the integral equality
p
t
0
sp−1|X|> sdst
p
|X|> t |X|
p1
|X|≤t, 2.8
it follows that
|Xn|
p1
|Xn|≤t≤p
t
0
sp−1Xn> sds
≤Cp
t
0
sp−1|X|> sdsC
tp|X|> t |X|
p1
|X|≤t.
Lemma 2.7cf.7. Let{An, n≥1}be a sequence of events defined onΩ,F, P. If∞n1PAn<
∞, thenPlim supnAn 0, if∞n1PAn ∞andPAk∩Am≤PAkPAmfork /m, then
Plim supnAn 1.
3. Strong Law of Large Numbers
Theorem 3.1. Letg·, h·, andφ·be as in Theorem1.1, and let{X, Xn, n ∈ } be a sequence
of negatively associated random variables with EXn 0. Assume that {Xn, n ∈ } ≺ X. If
Eφ−1|X|<∞, then
lim
n
1
gn
n
k1
Xk
hk0, a.s. 3.1
Conversely, let{X, Xn, n∈ }be a sequence of identically distributed NA random variables, if3.1
is true, thenEφ−1|X|<∞.
Proof. Assume that Eφ−1X < ∞. To prove 3.1by applying the Kronecker lemma, it
suffices to show that
the series
∞
k1
Xk
φk converges a.s. 3.2
Here, we shall use the three-series theorem for NA RVs.
LetYkXk/φk1|Xk/φk|≤1. Then, byEφ−1|X|<∞, we have
∞
k1
P Xk φk
>1
≤C
∞
k1
P X φk
>1
C
∞
k1
Pφ−1|X|> k≤CEφ−1|X|<∞,
3.3
which shows thatP{{|Xk/φk|>1}, i.o.}0, and
∞
k1
Xk
φk
1|Xk/φk|>1<∞ a.s. 3.4
Therefore, fromEXk0, it follows that
∞
k1
|EYk| ≤
∞
k1
E Xk φk
To this end we estimate the series
∞
k1
VarYk≤
∞
k1
EYk2E
∞
k1
X2 k
φ2k1|Xk/φk|≤1
≤C
∞
k1
E1|X|>φkE X
2
φ2k1|X/φk|≤1
≤C
∞
k1
P|X|> φk C
k0C
∞
kk01
EX2
φ2k11|X|≤φk
≤CEφ−1|X|Ck0CE
X2
∞
φ−1|X|
1
φ2xdx
≤CEφ−1|X|Ck0CaE
φ−1|X|Cb <∞.
3.6
Conversely, since{X, Xn, n∈ }are identically NA RVs. If3.1holds, that is,
1
φn
n
k1
Xko1, a.s. 3.7
It follows that
Xn
φn
n
k1Xk
φn −
φn−1
φn
n−1
k1Xk
φn−1 −→o1 a.s. 3.8
which shows thatφ−1nX
n → 0 a.s. Hence,
φ−1nX±n −→0 a.s., 3.9
wherexmax0, xandx−max0,−x.
Since{Xn±, n∈ }is still an NA sequence. Defining the following events,
An Xn>
εφn
3
, Bn X−n >
εφn
3
, 3.10
we havePAk∩Al≤PAkPAl, andPBk∩Bl≤PBkPBl, fork /l. By Lemma2.7, if
φ−1nX±
n → 0 a.s., then
∞
n1P{Xn>1/3εφn}<∞and
∞
n1P{Xn− >1/3εφn}<∞.
Therefore,
∞
n1
P|Xn|> εφn ≤
∞
n1
P Xn> 1
3εφn
∞
n1
P Xn−> 1
3εφn
<∞, ∀ε >0, 3.11
which is equivalent toEφ−1|X|<∞.
Theorem3.1also includes a particular case of logarithmic means, we can establish the following.
Corollary 3.2. Let{X, Xn, n∈ }be a sequence of NA RVs withEXn 0and{Xn, n∈ } ≺X. If
E|X|<∞, then, one has
lim
n
1 logn
n
k1
Xk
k 0, a.s. 3.12
Proof. Lethy y,gy logy, that is,φy ylogy. In this case,φ−1y ∼ y/logyas
y → ∞, thereforeE|X|α≤Eφ−1|X|≤E|X|, for 0< α <1.
Corollary 3.3. Let{X, Xn, n∈ }be a sequence of NA RVs withEXn0and{Xn, n∈ } ≺X. If
E|X|<∞, then, for everya, b≥0,ab >1/2, one has
lim
n
1
na n
k1
Xk
kb 0, a.s. 3.13
Remark 3.4. As pointed out by Jajte1, Theorem3.1includes several regular summability methods such as1the Kolmogorov SLLNhy 1, gy y;2the classical MZ SLLN
hy 1, gy y1/α,1≤α≤2.
4. Spitzer Type Law of Large Numbers
Since the definition of complete convergence was introduced by Hsu and Robins, there have been many authors who devote themselves to the study of the complete convergence for sums of independent and dependent RVs and obtain a series of elegant results, see4,8and reference therein.
We say that the Hsu-Robbins9law of large numbersLLNis valid if, for allε >0,
∞
n1
P{|Sn| ≥nε}<∞, 4.1
and the Spitzer10LLN is valid if, for allε >0,
∞
n1
1
nP{|Sn| ≥nε}<∞. 4.2
Theorem 4.1. Letφ·be as in Theorem1.1, and let{X, Xn, n ∈ } be a sequence of NA random
variables withEXn0. Assume that{Xn, n∈ } ≺X. IfEφ−1|X|<∞, then for allε >0,
∞
n1
1
nP
Conversely, let{X, Xn, n∈ }be a sequence of identically distributed NA random variables, if 4.3
is true, thenEφ−1|X|<∞andEX0.
Proof. For 1≤k≤n, letUknXk1|Xk|≤φn,V n
k Xk−U
n
k , then for everyn
|Sn|
n
k1
UknVkn≤
n
k1
Ukn−EUkn
n
k1
EUkn
n
k1
Vkn. 4.4
Note that
|Sn| ≥εφn ⊂
!
n
k1
Ukn−EUkn> εφn
2
"
∪!n
k1
Vkn> εφn
2
"
∪
!
1
φnmax1≤i≤n
n
k1
EUkn−→0
"
.
4.5
For the first term on the RHS of4.5, by Markov inequality and Lemma2.2and 3.6, we have
∞
n1
1 nP ! n
k1
Ukn−EUkn> εφn
2
"
≤∞
n1
C nφ2n
n
k1
EUkn2 SinceUkn, 1≤k≤nis also NA
C
∞
n1
1
nφ2n n
k1
EX2k1|Xk|≤φn
≤C
∞
n1
1
n
n
k1
E1|X|≥φnEX
21
|X|≤φn
φ2n
C
∞
n1
E1|X|≥φnEX
21
|X|≤φn φ2n
<∞.
4.6
For the second term on the RHS of4.5, since
!
n
k1
Vkn
> εφn2
"
⊂#n
k1
hence,
∞
n1
1
nP
!
n
k1
Vkn> εφn
2
"
≤∞
n1
1
n
n
k1
P|Xk|> φn
≤∞
n1
C n
n
k1
P|X|> φn
C
∞
n1
P|X|> φn <∞.
4.8
For the third term on the RHS of4.5, we have, by3.5,
1
φn
n
k1
EUkn 1
φn
n
k1
EVkn ByEXk0
n
k1EXk1|Xk|>φn φn
≤
n
k1E|Xk|1|Xk|>φn φn
≤CEφ−1|X|<∞.
4.9
Therefore,4.3follows.
Conversely, since ∞n11/nP{|Sn| ≥ εφn} < ∞ imply that Sn/φn o1a.s.,
hence from Theorem3.1, we haveEφ−1|X|<∞. These complete the proof of Theorem4.1.
Corollary 4.2. Under the assumptions of Theorem4.1, one has
∞
n1
1
nP 1max≤k≤n|Sk| ≥εφn
<∞. 4.10
Proof. DenoteSkn
k
i1U n
k , and noticing that
max
1≤k≤n|Sk| ≥εφn
⊂ max
1≤k≤n
Skn≥εφn
∪
!n
#
k1
|Xk|> φn
"
, 4.11
Analogously, we can prove the following corollaries, and omit the details.
Corollary 4.3. Let{X, Xn, n ∈ } be a sequence ofϕ-mixing random variables with EXn 0.
Assume that{Xn, n∈ } ≺X. IfEφ−1|X|<∞, and
∞
r1
ϕ1/2r<∞, 4.12
then4.3holds.
Corollary 4.4. Let{X, Xn, n∈ } be a sequence ofm-dependent random variables withEXn 0.
Assume that{Xn, n∈ } ≺X. IfEφ−1|X|<∞, then4.3holds.
Corollary 4.5. Let{X, Xn, n∈ }be a sequence of PA random variables withEXn0. Assume that
{Xn, n∈ } ≺X. IfEφ−1|X|<∞, and
∞
n1
μ
1
22n<∞, 4.13
then4.3holds.
5. Hsu-Robbins Type Law of Large Numbers
Theorem 5.1. Letφ·, be define as in Theorem1.1, but the following condition (3) is replaced by
3There exist constantsasuch thatφps∞
s xp/2/φpxdx≤as1p/2,s > d,p >2, and
let{X, Xn, n ∈ } be a sequence of NA random variables withEXn 0. Assume that
{Xn, n∈ } ≺X. IfEφ−1|X|1p/2<∞, then for allε >0,
∞
n1
P|Sn| ≥εφn <∞. 5.1
Proof. From the previous section, we know that to prove Theorem5.1, we need only to prove the convergence of the following three series.
First, note thatEφ−1|X|1p/2 <∞ ⇒Eφ−1|X|2 <∞ ⇒nP|X|> φn → 0, we
have
1
φn
n
k1
EUkn 1
φn
n
k1
EXk1|X
k|≤φn
1 φn
n
k1
EXk1|X
k|>φn SinceEXk0
≤ 1
φn
n
k1
φn1 n
k1
∞
0
P|Xk|1|Xk|>φn> t dt
≤ φnCn ·φnP|X|> φn φnCn
∞
n
Pφ−1|X|> tdt
≤CnP|X|> φn Cn φn
∞
n
Eφ−1|X|2 t2 dt
CnP|X|> φn CnE
φ−1|X|2
φn ·
1
n
CnP|X|> φn CE
φ−1|X|2
φn −→0.
5.2
Hence,1/φnmaxjjk1E|Ukn| → 0 asn → ∞. Next, by4.4, we have
∞
n1
P!
n
k1
Vkn> εφn
2
"
≤∞
n1 n
k1
P|Xk|> φn ≤
∞
n1
C
n
k1
P|X|> φn
≤C
∞
n1
nP|X|> φn < CEφ−1|X|2<∞.
5.3
Last, from the definition ofUknand the NA’s property, we know that{Ukn, 1≤k≤n, n≥1} remains a sequence of NA RVs. By applying Lemma2.2andCrinequality, we have
∞
n1
P!
n
k1
Ukn−EUkn> εφn
2
"
≤∞
n1
C φpnE
n
k1
Ukn−EUkn
p
∞
n1
C φpn
n
k1
EUkn−EUknp
∞
n1
C φpn
n
k1
EUkn−EUkn2
p/2
:I1I2.
5.4
It is easy to see that
I1≤C
∞
n1
1
φpn n
k1
EUknp
≤C
∞
n1
1
φpn
n
k1
φpnP|X|> φn
n
k1
E|X|p1
|X|≤φn
C
∞
n1
nP|X|> φn C
∞
n1
n φpnE|X|
p1
|X|≤φn
≤CEφ−1|X|21
2k0k01CC
∞
kk01
k1p/2
φpk1E|X| p1
|X|≤φk
≤CEφ−1|X|1p/2Ck2 0CE
|X|p
∞
φ−1|X|
xp/2
φpxdx
≤CEφ−1|X|1p/2Ck02<∞,
5.5
I2≤C
∞
n1
1
φpn
n
k1
EUkn2
p/2
≤C
∞
n1
1
φpn
nφ2nP|X|> φn nEX21|X|≤φn
p/2
≤C
∞
n1
1
φpn
φpnnp/2P|X|> φnC
∞
n1
np/2
φpn
EX21|X|≤φn
p/2
≤C
∞
n1
np/2P|X|> φn C
∞
n1
np/2
φpnE|X| p1
|X|≤φn
≤CEφ−1|X|1p/2<∞.
5.6
These complete the proof of Theorem5.1.
Acknowledgments
This work is supported by the National Nature Science Foundation of Chinano. 11071104 and the Anhui high Education Research Grantno. KJ2010A337. The author expresses his sincere gratitude to the referees and the editors for their hospitality.
References
1 R. Jajte, “On the strong law of large numbers,”The Annals of Probability, vol. 31, no. 1, pp. 409–412, 2003.
2 J. D. Esary, F. Proschan, and D. W. Walkup, “Association of random variables, with applications,” Annals of Mathematical Statistics, vol. 38, pp. 1466–1474, 1967.
3 K. Joag-Dev and F. Proschan, “Negative association of random variables, with applications,” The Annals of Statistics, vol. 11, no. 1, pp. 286–295, 1983.
4 C. Su and Y. B. Wang, “Strong convergence for identically distributed negatively associated sequences,” Chinese Journal of Applied Probability and Statistics, vol. 14, no. 2, pp. 131–140, 1998 Chinese.
6 S. C. Yang, “Moment ineqaulities for partial sums of random variables,”Science in China Series A, vol. 44, no. 3, pp. 218–223, 2003.
7 P. Matuła, “A note on the almost sure convergence of sums of negatively dependent random variables,”Statistics & Probability Letters, vol. 15, no. 3, pp. 209–213, 1992.
8 J. Bingyi and H. Y. Liang, “Strong limit theorems for weighted sums of negatively associated random sequence,”Journal of Theoretical Probability, vol. 21, no. 4, pp. 890–909, 2008.
9 P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,”Proceedings of the National Academy of Sciences of the United States of America, vol. 33, pp. 25–31, 1947.