-Mixing Random Variables

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,

Chang-Chun Tan,

and Jin-Guan Lin

1Department 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,[email protected] 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 corr

f, g

:∀S×T⊂N×N, distS, T≥n, ∀f∈L²FS, g∈L²FT

, 1.1

where

corr f, g

Cov

fXi; i∈S, g

Xj; j∈T Var

fXi; i∈S Var

g

X_j; j ∈T ^1/2. 1.2

Definition 1.1. A random variable sequence{Xn; n ≥ 1}is said to be a ρ^∗-mixing random variable sequence if there existsk∈Nsuch thatρ^∗_k<1.

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_∞

n1n^r−2P|_n

i1X_i|> n^1/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 0 forα≤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

n^αp−2P

max1≤j≤n

j i1

a_niX_i > εn^α

<∞. 1.5

Theorem C. Let{Xn; n≥1}be aρ^∗-mixing sequence of identically distributed random variables, αp > 1,α >1/2, andEX₁ 0 forα≤ 1. Assume that{ani; 1 ≤ i≤ n}is array of real numbers satisfying1.3. Then

n^−1/pn

i1

aniXi−→0 a.s.n−→ ∞. 1.6

Recently, Sung15obtained the following complete convergence results for weighted sums of identically distributed NA random variables.

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,nⁿ

i1

|ani|^α

n 1.7

for some 0< α≤2. Letbnn^1/αlogn^1/γfor someγ >0. Furthermore, suppose thatEX0 where 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

max1≤j≤n

j i1

a_niX_i > b_nε

<∞ ∀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,nⁿ

i1

|ani|^β

n , 1.10

whereβmaxα, γfor some 0< α≤2 andγ >0. Letbnn^1/αlogn^1/γ. IfEX0 for 1< 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 the caseαγholds forρ^∗-mixing sequence.

Theorem 1.4. Under the conditions ofTheorem 1.2, the assumptionsEX0 for 1< α≤2 and1.8 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 ≤rwithEX_i 0 andE|Xi|^q <

∞for everyi≥1, then for alln≥1,

E

max1≤i≤n|S_i|^q

≤D

⎛

⎝ⁿ

i1

E|X_i|^q _n

i1

EX²_{i q/2}⎞

⎠, 2.1

whereS_{i i}

j1X_j.

Lemma 2.2. Let X be a random variable and{ani; 1 ≤ i ≤ n, n ≥ 1} be an array of constants satisfying1.10,b_nn^1/αlogn^1/γ. Then

∞ n1

n⁻¹ⁿ

i1

P|aniX|> b_n≤

⎧⎪

⎨

⎪⎩

CE|X|^α forα > γ, CE|X|^γ forα < γ.

2.2

Proof. Ifγ > α, by_n

i1|ani|^γOnand Lyapounov’s inequality, then

1 n

n i1

|a_ni|^α≤ 1

n n

i1

|a_ni|^γ _α/γ

O1. 2.3

Hence,1.7is satisfied. From the proof of2.1of Sung15, we obtain easily that the result holds.

Proof ofTheorem 1.2. LetXnianiXiI|aniXi| ≤bn. For allε >0, we have ∞

n1

1 nP

max1≤j≤n

j i1

a_niX_i > εb_n

≤^∞

n1

1 nP

max1≤j≤na_njX_j> b_n

^∞

n1

1 nP

max1≤j≤n

j i1

X_ni > εb_n

:I₁I₂.

2.4 To obtain1.9, we need only to prove thatI1 <∞andI2<∞.

ByLemma 2.2, one gets

I₁≤^∞

n1

1 n

n j1

Pa_njX_j> b_n ^∞

n1

1 n

n j1

Pa_njX> b_n

<∞. 2.5

Before the proof ofI₂<∞, we prove firstly

b⁻¹_n max

1≤j≤n

j i1

EaniXiI|aniXi| ≤bn

−→0, asn−→ ∞. 2.6

For 0< α≤1,

b_n⁻¹max

1≤j≤n

j i1

Ea_niX_iI|aniX_i| ≤b_n

≤b⁻¹_n ⁿ

i1

E|aniX_i|I|aniX_i| ≤b_n≤b^−α_n ⁿ

i1

|ani|^αE|X|^α

≤C

logn_−α/γ

E|X|^α−→0, asn−→ ∞.

2.7

For 1< α≤2,

b_n⁻¹max

1≤j≤n

j i1

EaniXiI|aniXi| ≤bn

b⁻¹_n max

1≤j≤n

j i1

EaniXiI|aniXi|> bn

EXi0

≤b⁻¹_n ⁿ

i1

E|aniXi|I|aniXi|> bn≤b^−α_n ⁿ

i1

|ani|^αE|X|^α

≤C

logn_−α/γ

E|X|^α−→0, asn−→ ∞.

2.8

Thus2.6holds. So, to proveI₂<∞, it is enough to show that

I3^∞

n1

1 nP

max1≤j≤n

j i1

Xni−EXni

> εbn

<∞, ∀ε >0. 2.9

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

X_ni−EX_ni

q⎞

⎠

≤C^∞

n1

n⁻¹b^−q_n ⁿ

i1

E|aniXi|^qI|aniXi| ≤bn

C^∞

n1

n⁻¹b^−q_{n n}

i1

Ea_niX_i²I|a_niX_i| ≤b_{n 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 α

logn₋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⁻¹

logn_−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_n^−αq/2 _n

i1

|ani|^αE|X|^α q/2

≤C^∞

n1

n⁻¹

logn_−αq/2γ

<∞.

2.13

Ifγ > 2 ≥ αorγ ≥ 2 > α, one getsE|X|² < ∞. Since_n

i1|ani|^α On, it implies max_1≤i≤n|ani|^α≤Cn. Therefore, we have

n i1

|ani|^kn

i1

|ani|^α|ani|^k−α≤Cnn^k−α/α Cn^k/α 2.14

for allk≥α. Hence,_n

i1|ani|²On^2/α. Takingq > γ, we have

I32 ≤C^∞

n1

n⁻¹b^−qn

_n

i1

|ani|² _q/2

≤C^∞

n1

n⁻¹b^−q_n n^q/αC^∞

n1

n⁻¹

logn_−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

max_1≤j≤2ⁱj

i1a_niX_i 2^i1/α

log 2ⁱ¹1/γ −→0 a.s.i−→ ∞. 2.16

For anyn≥1, there exists an integerisuch that 2ⁱ⁻¹≤n <2ⁱ. So

2ⁱ⁻¹max≤n<2ⁱ

_n

j1a_njX_j

b_n ≤ max_1≤j≤2ⁱj

i1a_njX_j 2^i−1/α

log 2ⁱ⁻¹_1/γ 2^2/α

max_1≤j≤2ⁱ_n

j1a_njX_j 2^i1/α

log 2ⁱ¹_1/γ

i1 i−1

1/γ

. 2.17 From2.16and2.17, we have

nlim→ ∞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.

References

1 R. C. Bradley, “On the spectral density and asymptotic normality of weakly dependent random fields,” Journal of Theoretical Probability, vol. 5, no. 2, pp. 355–373, 1992.

2 W. Bryc and W. Smole ´nski, “Moment conditions for almost sure convergence of weakly correlated random variables,” Proceedings of the American Mathematical Society, vol. 119, no. 2, pp. 629–635, 1993.

3 S. C. Yang, “Some moment inequalities for partial sums of random variables and their application,”

Chinese Science Bulletin, vol. 43, no. 17, pp. 1823–1828, 1998.

4 Q. Y. Wu, “Convergence for weighted sums ofρmixing random sequences,” Mathematica Applicata, vol. 15, no. 1, pp. 1–4, 2002Chinese.

5 Q. Wu and Y. Jiang, “Some strong limit theorems forρ-mixing sequences of random variables,”

Statistics & Probability Letters, vol. 78, no. 8, pp. 1017–1023, 2008.

6 M. Peligrad and A. Gut, “Almost-sure results for a class of dependent random variables,” Journal of Theoretical Probability, vol. 12, no. 1, pp. 87–104, 1999.

7 S. X. Gan, “Almost sure convergence forρ-mixing random variable sequences,” Statistics & Probability Letters, vol. 67, no. 4, pp. 289–298, 2004.

8 S. Utev and M. Peligrad, “Maximal inequalities and an invariance principle for a class of weakly dependent random variables,” Journal of Theoretical Probability, vol. 16, no. 1, pp. 101–115, 2003.

9 J. An and D. M. Yuan, “Complete convergence of weighted sums forρ^∗-mixing sequence of random variables,” Statistics & Probability Letters, vol. 78, no. 12, pp. 1466–1472, 2008.

10 K. Budsaba, P. Chen, and A. Volodin, “Limiting behaviour of moving average processes based on a sequence ofρ⁻ mixing and negatively associated random variables,” Lobachevskii Journal of Mathematics, vol. 26, pp. 17–25, 2007.

11 L E. Baum and M. Katz, “Convergence rates in the law of large numbers,” Transactions of the American Mathematical Society, vol. 120, pp. 108–123, 1965.

12 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.

13 P. Erd ¨os, “On a theorem of Hsu and Robbins,” Annals of Mathematical Statistics, vol. 20, pp. 286–291, 1949.

14 F. Spitzer, “A combinatorial lemma and its application to probability theory,” Transactions of the American Mathematical Society, vol. 82, pp. 323–339, 1956.

15 S. H. Sung, “On the strong convergence for weighted sumsof random variables,” Statistical Papers. In press.

16 P. Chen, T.-C. Hu, X. Liu, and A. Volodin, “On complete convergence for arrays of rowwise negatively associated random variables,” Rossi˘ıskaya Akademiya Nauk, vol. 52, no. 2, pp. 393–397, 2007.