A Note on the Almost Sure Central Limit Theorem for Partial Sums of ρ⁻-Mixing Sequences

Feng Xu, Qunying Wu

College of Science, Guilin University of Technology, Guilin, China.

DOI: 10.4236/am.2015.69140                

Abstract

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.

Keywords

ρ⁻-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. [1] A sequence is called ρ⁻-mixing, if

where

The definition of NA is given by Joag-Dev and Proschan [2] , and the concept of ρ^*-mixing random variables is given by Kolmogorov and Rozanov [3] . In 1999, the concept of ρ⁻-mixing random variables was introduced initially by Zhang and Wang [1] . 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 [1] for Rosenthal-type moment inequality and Marcinkiewicz-Zygmund law of large numbers, Zhang [4] for the central limit theorems of random fields, Wang and Lu [5] for the weak convergence theorems.

Starting with Brosamler [6] and Schatte [7] , 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 [6] , Schatte [7] , Lacey and Philipp [8] , Ibragimov and Lifshits [9] , Berkes and Csáki [10] , Hörmann [11] and Wu [12] . The simplest form of the ASCLT [6] - [8] reads as follows: let be i.i.d. random variables with mean 0, variance and partial sums. Then

(1)

where I denotes indicator function, and is the standard normal distribution function. For other version of ρ⁻-mixing sequences, see [13] -[15] .

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 [6] , Schatte [7] and Lacey and Philipp [8] 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

(a)

(b)

(c)

Suppose and set

(2)

then

(3)

Remark 1. By the terminology of summation procedures (cf. [16] , 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. [4] Let be a weakly stationary ρ⁻-mixing sequence with, , and

, , then

where denotes the standard normal random variable.

Lemma 2. [5] 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. [17] 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

(4)

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 [18] , 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 [19] and Section 2 of Peligrad and Shao [20] , we know that (3) is equivalent to

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

(5)

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

Let, define

For any, we get,

(6)

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

(7)

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

(8)

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).

Conflicts of Interest

The authors declare no conflicts of interest.

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