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

Let { } n n N X ∈ 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 n n S σ , where ∑ n n i i S X 1 = = , n n S σ 2 2 E =. The result generalizes and improves the previous results.


Introduction
Let C be a class of functions which are coordinatewise increasing.For a random variable X, define ( ) For two nonempty disjoint sets , S T N ⊂ , we define 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.
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 1 k to ( ) ; and set the positive absolute constant c to vary from line to line.

Some Lemmas
, as , where  denotes the standard normal random variable.Lemma 2. [5] For a positive real number be a weakly stationary ρ − -mixing sequence.Assume sup E .
, n n N ξ ξ ∈ be a sequence of uniformly bounded random variables.Assume that ( ) where k d and n D are defined by (2). .
Firstly we estimate 1 n T .Since k ξ is a bounded random variable, we get Now we estimate 2 n T .By the conditions ( ) exp log exp log .
exp log exp log exp log exp log exp log .
Wu [18], we have, as n → ∞ , ( ) For any n, existing k n and , then, by i c ξ ≤ for any i, . i.e., (4) holds.This completes the proof of Lemma 4.

Proof
Proof of Theorem 1.By Lemma 1, we have , as .
This implies that for any ( ) g x 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 ( ) for any ( ) For any 1 2k l ≤ < , we get, Firstly we estimate 1 I .By Lemma 1 x y ∈ .By Jensen inequality, Lemma 2 and σ < ∞ , we obtain that ( ) ( ) ( ) Now we estimate 2 I .Note that g is a bounded function with bounded continuous derivatives, so, by Lemma So if 2 l k > , combining with ( 6), (7), (8), we obtain ( ) By Lemma 4, (5) holds.This completes the proof of Theorem 1.1.
associated (NA) if for ever pair of disjoint subsets S, T of N,
Then for any bounded Lipschitz function f: R R → , We have g is a bounded Lipschitz function, i.e., there exists a constant c > 0 such that