Volume 2011, Article ID 157816,8pages doi:10.1155/2011/157816
Research Article
On the Strong Laws for Weighted Sums of
ρ
∗
-Mixing Random Variables
Xing-Cai Zhou,
1, 2Chang-Chun Tan,
3and Jin-Guan Lin
11Department of Mathematics, Southeast University, Nanjing 210096, China
2Department of Mathematics and Computer Science, Tongling University, Tongling, Anhui 244000, China
3School of Mathematics, Heifei University of Technology, Hefei, Anhui 230009, China
Correspondence should be addressed to Chang-Chun Tan,cctan@ustc.edu.cn
Received 26 October 2010; Revised 5 January 2011; Accepted 27 January 2011
Academic Editor: Matti K. Vuorinen
Copyrightq2011 Xing-Cai Zhou et al. 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.
Complete convergence is studied for linear statistics that are weighted sums of identically distributedρ∗-mixing random variables under a suitable moment condition. The results obtained generalize and complement some earlier results. A Marcinkiewicz-Zygmund-type strong law is also obtained.
1. Introduction
Suppose that{Xn; n≥ 1}is a sequence of random variables andSis a subset of the natural number setN. LetFSσXi; i∈S,
ρ∗
nsup
corrf, g:∀S×T⊂N×N, distS, T≥n, ∀f∈L2F
S, g∈L2FT
, 1.1
where
corrf, g Cov
fXi; i∈S, gXj; j∈T
VarfXi; i∈SVargXj; j ∈T 1/2
. 1.2
The notion of ρ∗-mixing seems to be similar to the notion of ρ-mixing, but they are quite different from each other. Many useful results have been obtained forρ∗-mixing
random variables. For example, Bradley1has established the central limit theorem, Byrc and Smole ´nski 2 and Yang 3 have obtained moment inequalities and the strong law of large numbers, Wu4,5, Peligrad and Gut 6, and Gan7have studied almost sure convergence, Utev and Peligrad8have established maximal inequalities and the invariance principle, An and Yuan9have considered the complete convergence and Marcinkiewicz-Zygmund-type strong law of large numbers, and Budsaba et al.10have proved the rate of convergence and strong law of large numbers for partial sums of moving average processes based onρ−-mixing random variables under some moment conditions.
For a sequence{Xn; n ≥ 1} of i.i.d. random variables, Baum and Katz 11proved the following well-known complete convergence theorem: suppose that {Xn; n ≥ 1} is a sequence of i.i.d. random variables. ThenEX1 0 andE|X1|rp <∞1≤ p <2, r ≥1if and only if∞n1nr−2P|n
i1Xi|> n1/pε<∞for allε >0.
Hsu and Robbins12and Erd ¨os13proved the case r 2 andp 1 of the above theorem. The caser 1 andp1 of the above theorem was proved by Spitzer14. An and
Yuan9studied the weighted sums of identically distributedρ∗-mixing sequence and have the following results.
Theorem B. Let{Xn; n≥ 1}be aρ∗-mixing sequence of identically distributed random variables,
αp >1,α >1/2, and suppose thatEX1 0forα≤1. Assume that{ani; 1≤i≤n}is an array of real numbers satisfying
n
i1
|ani|pOδ, 0< δ <1, 1.3
Ank
1≤i≤n:|ani|p>k1−1
≥ne−1/k. 1.4
IfE|X1|p<∞, then
∞
n1 nαp−2P
max 1≤j≤n
j
i1 aniXi
> εnα
<∞. 1.5
Theorem C. Let{Xn; n≥1}be aρ∗-mixing sequence of identically distributed random variables,
αp > 1,α >1/2, andEX1 0forα≤ 1. Assume that{ani; 1 ≤ i≤ n}is array of real numbers
satisfying1.3. Then
n−1/pn i1
aniXi−→0a.s.n−→ ∞. 1.6
Theorem D. Let{X, Xn; n≥1}be a sequence of identically distributed NA random variables, and let{ani; 1≤i≤n, n≥1}be an array of constants satisfying
Aαlim sup
n→ ∞ Aα,n<∞, Aα,n
n
i1
|ani|α
n 1.7
for some0< α≤2. Letbnn1/αlogn1/γfor someγ >0. Furthermore, suppose thatEX0where 1< α≤2. If
E|X|α<∞, forα > γ,
E|X|αlog|X|<∞, forαγ,
E|X|γ<∞, forα < γ,
1.8
then
∞
n1 1 nP
max 1≤j≤n
j
i1 aniXi
> bnε
<∞ ∀ε >0. 1.9
We find that the proof of Theorem C is mistakenly based on the fact that1.5holds for αp 1. Hence, the Marcinkiewicz-Zygmund-type strong laws forρ∗-mixing sequence have not been established.
In this paper, we shall not only partially generalize Theorem D toρ∗-mixing case, but also extend Theorem B to the caseαp1. The main purpose is to establish the Marcinkiewicz-Zygmund strong laws for linear statistics ofρ∗-mixing random variables under some suitable conditions.
We have the following results.
Theorem 1.2. Let{X, Xn;n≥1}be a sequence of identically distributedρ∗-mixing random variables, and let{ani; 1≤i≤n, n≥1}be an array of constants satisfying
Aβlim sup
n→ ∞ Aβ,n<∞, Aβ,n
n
i1
|ani|β
n , 1.10
whereβmaxα, γfor some0< α≤2andγ >0. Letbnn1/αlogn1/γ. IfEX0for1< α≤2 and1.8forα /γ, then1.9holds.
Remark 1.3. The proof of Theorem D was based on Theorem 1 of Chen et al.16, which gave
sufficient conditions about complete convergence for NA random variables. So far, it is not known whether the result of Chen et al.16holds forρ∗-mixing sequence. Hence, we use different methods from those of Sung15. We only extend the caseα /γ of Theorem D to ρ∗-mixing random variables. It is still open question whether the result of Theorem D about
Theorem 1.4. Under the conditions ofTheorem 1.2, the assumptionsEX0for1< α≤2and1.8
forα /γimply the following Marcinkiewicz-Zygmund strong law:
b−1
n n
i1
aniXi −→0 a.s.n−→ ∞. 1.11
2. Proof of the Main Result
Throughout this paper, the symbolCrepresents a positive constant though its value may change from one appearance to next. It proves convenient to define logx max1,lnx, where lnxdenotes the natural logarithm.
To obtain our results, the following lemmas are needed.
Lemma 2.1Utev and Peligrad8. SupposeNis a positive integer,0≤r <1, andq≥2. Then there exists a positive constantDDN, r, qsuch that the following statement holds.
If{Xi; i≥1}is a sequence of random variables such thatρ∗N ≤rwithEXi 0andE|Xi|q <
∞for everyi≥1, then for alln≥1,
Emax 1≤i≤n|Si|
q≤D ⎛
⎝n
i1
E|Xi|q
n
i1 EX2
i q/2⎞
⎠, 2.1
whereSi ij1Xj.
Lemma 2.2. Let X be a random variable and{ani; 1 ≤ i ≤ n, n ≥ 1} be an array of constants satisfying1.10,bnn1/αlogn1/γ. Then
∞
n1 n−1n
i1
P|aniX|> bn≤ ⎧ ⎪ ⎨ ⎪ ⎩
CE|X|α forα > γ,
CE|X|γ forα < γ. 2.2
Proof. Ifγ > α, byni1|ani|γOnand Lyapounov’s inequality, then
1 n
n
i1
|ani|α≤
1 n
n
i1
|ani|γ α/γ
O1. 2.3
Hence,1.7is satisfied. From the proof of2.1of Sung15, we obtain easily that the result
Proof ofTheorem 1.2. LetXnianiXiI|aniXi| ≤bn. For allε >0, we have ∞ n1 1 nP max 1≤j≤n
j i1 aniXi
> εbn
≤∞ n1 1 nP max 1≤j≤n
anjXj> bn ∞ n1 1 nP max 1≤j≤n
j i1 Xni > εbn
:I1I2.
2.4
To obtain1.9, we need only to prove thatI1 <∞andI2<∞. ByLemma 2.2, one gets
I1≤
∞ n1 1 n n j1
PanjXj> bn
∞ n1 1 n n j1
PanjX> bn<∞. 2.5
Before the proof ofI2<∞, we prove firstly
b−1
n max1≤j≤n j i1
EaniXiI|aniXi| ≤bn
−→0, asn−→ ∞. 2.6
For 0< α≤1,
b−1
n max
1≤j≤n j i1
EaniXiI|aniXi| ≤bn ≤b−n1
n
i1
E|aniXi|I|aniXi| ≤bn≤b−nα n
i1
|ani|αE|X|α
≤Clogn−α/γE|X|α−→0, asn−→ ∞.
2.7
For 1< α≤2,
b−1
n max1≤j≤n j i1
EaniXiI|aniXi| ≤bn
b−n1max1≤j≤n j i1
EaniXiI|aniXi|> bn
EXi0
≤b−1
n n
i1
E|aniXi|I|aniXi|> bn≤b−nα n
i1
|ani|αE|X|α
≤Clogn−α/γE|X|α−→0, asn−→ ∞.
2.8
Thus2.6holds. So, to proveI2<∞, it is enough to show that
I3∞ n1 1 nP
max 1≤j≤n j i1
Xni−EXni > εbn
By the Chebyshev inequality andLemma 2.1, forq≥max{2, γ}, we have
I3≤C
∞
n1 n−1b−q
n E ⎛
⎝max
1≤j≤n
j
i1
Xni−EXni
q⎞ ⎠
≤C∞
n1 n−1b−q
n n
i1
E|aniXi|qI|aniXi| ≤bn
C∞
n1 n−1b−q
n
n
i1
EaniXi2I|aniXi| ≤bn q/2
:I31I32.
2.10
ForI31, we consider the following two cases.
Ifα < γ, note thatE|X|γ <∞. We have
I31≤C∞
n1 n−1b−γ
n n
i1
|ani|γE|X|γ ≤C
∞
n1 n−
γ
αlogn−1<∞. 2.11
Ifα > γ, note thatE|X|α<∞. we have
I31 ≤C
∞
n1 n−1b−α
n n
i1
|ani|αE|X|α≤C
∞
n1
n−1logn−α/γ<∞. 2.12
Next, we proveI32<∞in the following two cases.
Ifα < γ≤2 orγ < α≤2, takeq >max2,2γ/α. Noting thatE|X|α<∞, we have
I32≤C∞
n1
n−1b−αq/2
n
n
i1
|ani|αE|X|α q/2
≤C∞
n1
n−1logn−αq/2γ<∞.
2.13
Ifγ > 2 ≥ αorγ ≥ 2 > α, one getsE|X|2 < ∞. Sincen
i1|ani|α On, it implies max1≤i≤n|ani|α≤Cn. Therefore, we have
n
i1
|ani|k n
i1
for allk≥α. Hence,ni1|ani|2On2/α. Takingq > γ, we have
I32 ≤C∞
n1 n−1b−q
n
n
i1
|ani|2 q/2
≤C∞
n1 n−1b−q
n nq/αC
∞
n1
n−1logn−q/γ <∞.
2.15
Proof ofTheorem 1.4. By1.9, a standard computationsee page 120 of Baum and Katz11
or page 1472 of An and Yuan9, and the Borel-Cantelli Lemma, we have
max1≤j≤2ij
i1aniXi
2i1/αlog 2i11/γ −→0 a.s.i−→ ∞. 2.16
For anyn≥1, there exists an integerisuch that 2i−1≤n <2i. So
max 2i−1≤n<2i
nj1anjXj bn ≤
max1≤j≤2iji
1anjXj 2i−1/αlog 2i−11/γ 2
2/αmax1≤j≤2i
nj1anjXj
2i1/αlog 2i11/γ
i
1 i−1
1/γ .
2.17
From2.16and2.17, we have
lim n→ ∞b
−1
n n
i1
aniXi 0 a.s. 2.18
Acknowledgments
The authors thank the Academic Editor and the reviewers for comments that greatly improved the paper. This work is partially supported by Anhui Provincial Natural Science Foundation no. 11040606M04, Major Programs Foundation of Ministry of Education of China no. 309017, National Important Special Project on Science and Technology 2008ZX10005-013, and National Natural Science Foundation of China11001052, 10971097, and 10871001.
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