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