^{1}

^{*}

^{1}

^{*}

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.

Let

For two nonempty disjoint sets

A sequence

where^{*}-mixing, if

where

Definition 1. [^{−}-mixing, if

where

The definition of NA is given by Joag-Dev and Proschan [^{*}-mixing random variables is given by Kolmogorov and Rozanov [^{−}-mixing random variables was introduced initially by Zhang and Wang [^{−}-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 [

Starting with Brosamler [

where I denotes indicator function, and ^{−}-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 [^{−}-mixing sequences, but also expand the scope of the weights from

Throughout this paper,

Theorem 1. Let ^{−}-mixing sequence with

Suppose

then

Remark 1. By the terminology of summation procedures (cf. [

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

Lemma 1. [^{−}-mixing sequence with

where

Lemma 2. [^{−}-mixing random variables with

Lemma 3. [^{−}-mixing sequence. Assume

Lemma 4. Let

then

where

Proof. Set

Firstly we estimate

Now we estimate

By condition

and

Since

Wu [

Thus

Let

By Borel-Cantelli lemma,

For any n, existing

from

Proof of Theorem 1. By Lemma 1, we have

This implies that for any

Hence, by the Toeplitz lemma, we obtain

In the other hand, from Theorem 7.1 of Billingsley [

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

for any

Let

For any

Firstly we estimate

Now we estimate

So if

By Lemma 4, (5) holds.

This completes the proof of Theorem 1.1.

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

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