AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2015.69140AM-58951ArticlesPhysics&Mathematics A Note on the Almost Sure Central Limit Theorem for Partial Sums of <i>ρ</i><sup>&minus;</sup>-Mixing Sequences engXu1*QunyingWu1*College of Science, Guilin University of Technology, Guilin, China* E-mail:xufeng34@163.com(EX);wqy666@glut.edu.cn(QW);0508201506091574158027 July 2015accepted 18 August 21 August 2015© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Let be a strictly stationary sequence of ρ?-mixing random variables. We proved the almost sure central limit theorem, containing the general weight sequences, for the partial sums , where , . The result generalizes and improves the previous results.

<i>ρ</i><sup>&minus;</sup>-Mixing Sequences Partial Sums Almost Sure Central Limit Theorem
1. Introduction

Let be a class of functions which are coordinatewise increasing. For a random variable X, define

.

For two nonempty disjoint sets, we define to be. Let be the -field generated by, and define similarly.

A sequence is called negatively associated (NA) if for ever pair of disjoint subsets S, T of N,

where. is called ρ*-mixing, if

where

Definition 1.  A sequence is called ρ-mixing, if

where

The definition of NA is given by Joag-Dev and Proschan  , and the concept of ρ*-mixing random variables is given by Kolmogorov and Rozanov  . In 1999, the concept of ρ-mixing random variables was introduced initially by Zhang and Wang  . Obviously, ρ-mixing random variables include NA and ρ*-mixing random variables, which have a lot of applications. Their limit properties have received more and more attention recently, and a number of results have been obtained, such as Zhang and Wang  for Rosenthal-type moment inequality and Marcinkiewicz-Zygmund law of large numbers, Zhang  for the central limit theorems of random fields, Wang and Lu  for the weak convergence theorems.

Starting with Brosamler  and Schatte  , in the last two decades several authors investigated the almost sure central limit theorem (ASCLT) for partial sums of random variables. We refer the reader to Brosamler  , Schatte  , Lacey and Philipp  , Ibragimov and Lifshits  , Berkes and Csáki  , Hörmann  and Wu  . The simplest form of the ASCLT  -  reads as follows: let be i.i.d. random variables with mean 0, variance and partial sums. Then

where I denotes indicator function, and is the standard normal distribution function. For other version of ρ-mixing sequences, see  - .

The purpose of this article is to study and establish the ASCLT, containing the general weight sequences, for partial sums of ρ-mixing sequence. Our results not only generalize and improve those on ASCLT previously obtained by Brosamler  , Schatte  and Lacey and Philipp  from the i.i.d. case to ρ-mixing sequences, but also expand the scope of the weights from to,.

Throughout this paper, means; and set the positive absolute constant c to vary from line to line.

Theorem 1. Let be a strictly stationary ρ-mixing sequence with, for a certain, and denote,. Assume that

Suppose and set

then

Remark 1. By the terminology of summation procedures (cf.  , p. 35), Theorem 1 remains valid if we replace the weight sequence by any such that and.

Remark 2. ρ-mixing random variables include NA and ρ*-mixing random variables, so for NA and ρ*-mixing random variables sequences Theorem 1 also holds.

Remark 3. Essentially, the open problem that whether Theorem 1 holds for still remains open.

2. Some Lemmas

Lemma 1.  Let be a weakly stationary ρ-mixing sequence with, , and

, , then

where denotes the standard normal random variable.

Lemma 2.  For a positive real number, if is a sequence of ρ-mixing random variables with, for every, then for all, there is a positive constant such that

Lemma 3.  Let be a weakly stationary ρ-mixing sequence. Assume Then for any bounded Lipschitz function f:, We have

Lemma 4. Let be a sequence of uniformly bounded random variables. Assume that

and existing constants and such that

then

where and are defined by (2).

Proof. Set, we get

Firstly we estimate. Since is a bounded random variable, we get

Now we estimate. By the conditions for, we get

By condition, we obtain

and

Since and for from the proof of Lemma 2.2 in

Wu  , we have, as,

Thus

Let, , we get

By Borel-Cantelli lemma,

For any n, existing and such that, then, by for any i,

from. i.e., (4) holds. This completes the proof of Lemma 4.

3. Proof

Proof of Theorem 1. By Lemma 1, we have

This implies that for any which is a bounded function with bounded continuous derivatives,

Hence, by the Toeplitz lemma, we obtain

In the other hand, from Theorem 7.1 of Billingsley  and Section 2 of Peligrad and Shao  , we know that (3) is equivalent to

Hence, to prove (3), it suffices to prove

for any which is a bounded function with bounded continuous derivatives.

Let, define

For any, we get,

Firstly we estimate. By Lemma 1, we note that certain, exist such that. Since g is a bounded Lipschitz function, i.e., there exists a constant c > 0 such that

, for any. By Jensen inequality, Lemma 2 and, we obtain that

Now we estimate. Note that g is a bounded function with bounded continuous derivatives, so, by Lemma 3, we have

So if, combining with (6), (7), (8), we obtain

By Lemma 4, (5) holds.

This completes the proof of Theorem 1.1.

Acknowledgments

We thank the editor and the referee for their comments. This work is supported by National Natural Science Foundation of China (11361019).

Cite this paper

FengXu,QunyingWu, (2015) A Note on the Almost Sure Central Limit Theorem for Partial Sums of ρ-Mixing Sequences. Applied Mathematics,06,1574-1580. doi: 10.4236/am.2015.69140

ReferencesZhang, L.X. and Wang, X.Y. (1999) Convergence Rates in the Strong Laws of Asymptotically Negatively Associated Random Fields. Applied Mathematics—A Journal of Chinese Universities Series B, 14, 406-416. http://dx.doi.org/10.1007/s11766-999-0070-6Joag-Dev, K. and Proschan, F. (1983) Negative Association of Random Variables with Applications. Annals of Statistics, 11, 286-295. http://dx.doi.org/10.1214/aos/1176346079Kolmogorov, A.N. and Rozanov, U.A. (1960) On Strong Mixing Conditions for Stationary Gaussian Processes. Theory of Probability and Its Applications, 5, 204-208. http://dx.doi.org/10.1137/1105018Zhang, L.X. (2000) Central Limit Theorems for Asymptotically Negatively Associated Random Fields. Acta Mathematica Sinica, 6, 691-710. http://dx.doi.org/10.1007/s101140000084Wang, J.F. and Lu, F.B. (2006) Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables. Acta Mathematica Sinica, 22, 693-700. http://dx.doi.org/10.1007/s10114-005-0601-xBrosamler, G.A. (1988) An Almost Everywhere Central Limit Theorem. Mathematical Proceedings of the Cambridge Philosophical Society, 104, 561-574. http://dx.doi.org/10.1017/S0305004100065750Schatte, P. (1988) On Strong Versions of the Central Limit Theorem. Mathematische Nachrichten, 137, 249-256. http://dx.doi.org/10.1002/mana.19881370117Lacey, M.T. and Philipp, W. (1990) A Note on the Almost Sure Central Limit Theorem. Statistics and Probability Letters, 9, 201-205. http://dx.doi.org/10.1016/0167-7152(90)90056-DIbragimov, I.A. and Lifshits, M. (1998) On the Convergence of Generalized Moments in Almost Sure Central Limit Theorem. Statistics and Probability Letters, 40, 343-351. http://dx.doi.org/10.1016/S0167-7152(98)00134-5Berkes, I. and Csáki, E. (2001) A Universal Result in Almost Sure Central Limit Theory. Stochastic Processes and Their Applications, 94, 105-134. http://dx.doi.org/10.1016/S0304-4149(01)00078-3H&ouml;rmann, S. (2007) Critical Behavior in Almost Sure Central Limit Theory. Journal of Theoretical Probability, 20, 613-636. http://dx.doi.org/10.1007/s10959-007-0080-3Wu, Q.Y. (2011) Almost Sure Limit Theorems for Stable Distribution. Statistics and Probability Letters, 281, 662-672. http://dx.doi.org/10.1016/j.spl.2011.02.003Zhang, M.D., Tan, X.L. and Zhang, Y. (2015) An Extension of Almost Sure Central Limit Theorem for Product of Partial Sums of ρ&minus;-Mixing Sequences. Journal of Beihua University (Natural Science), 16, 427-430.Tan, L.X., Zhang, Y. and Zhang, Y. (2012) An Almost Sure Central Limit Theorem of Products of Partial Sums for ρ&minus;-Mixing Sequences. Journal of Inequalities and Applications, 2012, 51-63.Zhou, G.Y. and Zhang, Y. (2014) Almost Sure Central Limit Theorem of Products of Sums of Partial Sums for ρ&minus;-Mixing Sequences. Journal of Jilin University (Science Edition), 50, 1129-1134.Chandrasekharan, K. and Minakshisundaram, S. (1952) Typical Means. Oxford University Press, Oxford.Zhou H. ,et al. (2005)A Note on the Almost Sure Central Limit Theorem for ρ&minus;-Mixing Sequences Chinese Journal Zhejiang University (Science Edition) 32, C503-C505.Wu Q.Y. ,et al. (2012)A Note on the Almost Sure Limit Theorem for Self-Normalized Partial Sums of Random Variables in the Domain of Attraction of Thenormal Law Journal of Inequalities and Applications 2012, 17-26.Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Peligrad, M. and Shao, Q.M. (1995) A Note on the Almost Sure Central Limit Theorem for Weakly Dependent Random Variables. Statistics and Probability Letters, 22, 131-136. http://dx.doi.org/10.1016/0167-7152(94)00059-H