1. Introduction
Let 
be a probability space. The random variables we deal with are all defined on
. Let 
be a sequence of random variables. For each nonempty set
, write
. Given 
-algebras 
in
, let
where
. Define the 
-mixing coefficients by
(1.1)
where (for a given positive integer
) this sup is taken over all pairs of nonempty finite subsets 
such that dist
.
Obviously 
and 
except in the trivial case where all of the random variables 
are degenerate.
Definition 1.1. A sequence of random variables 
is said to be a 
-mixing sequence of random variables if there exists 
such that
.
Without loss of generality we may assume that 
is such that 
(see [1]). Here we give two examples of the practical application of 
- mixing.
Example 1.1. According to the proof of Theorem 2 in [2] and Remark 3 in [1], if 
is a strictly stationary Gaussian sequence which has a bounded positive spectral density
, then the sequence
has the property that
. Therefore, instantaneous functions 
of such a sequence provides a class of examples for 
-mixing sequences.
Example 1.2. If 
has a bounded positive spectral density
, i.e., 
for every t, then
. Thus, 
is a 
-mixing sequence.
-mixing is similar to 
-mixing, but both are quite different. 
is defined by (1.1) with index sets restricted to subsets S of 
and subsets 
of
. On the other hand, 
-mixing sequence assume condition
，but 
-mixing sequence assume condition that there exists 
such that
, from this point of view, 
-mixing is weaker than 
-mixing.
A number of writers have studied 
-mixing sequences of random variables and a series of useful results have been established. We refer to [2] for the central limit theorem [1,3], for moment inequalities and the strong law of large numbers [4-9], for almost sure convergence, and [10] for maximal inequalities and the invariance principle. When these are compared with the corresponding results for sequences of independent random variables, there still remains much to be desired.
The main purpose of this paper is to study the complete convergence and weak law of large numbers of partial sums of 
-mixing sequences of random variables and try to obtain some new results. We establish the complete convergence theorems and the weak law of large numbers. Our results in this paper extend and improve the corresponding results of Feller [11] and Baum and Katz [12].
Lemma 1.1. ([10], Theorem 2.1) Suppose K is a positive integer, 
, and
. Then there exists a positive constant 
such that the following statement holds:
If 
is a sequence of random variables such that 
and 
and 
for all
, then for every
,
where
.
Lemma 1.2. Let 
be a 
-mixing sequence of random variables. Then for any
, there exists a positive constant c such that for all
,
Proof. Let 
and
. Without loss of generality, assume that
. By the Cauchy-Schwarz inequality and Lemma 1.2,
Thus
i.e.,
2. Complete Convergence
In the following, let 
denote
, and 
denote that there exists a constant 
such that 
for sufficiently large n, logx mean
, and
.
Definition 2.1. A measurable function 
is said to be a slowly varying function at 
if for any
,
.
Lemma 2.1 ([13], Lemma 1). Let 
be a slowly varying function at
. Then i)
.
ii) 
for any
.
iii) For any 
and
, there exist positive constants 
and 
(depending only on
, and the function
) such that for any positive number k,
iv) For any 
and
, there exist positive constants 
and 
(depending only on
, and the function
) such that for any positive number k,
Theorem 2.1. Let 
be a 
-mixing sequence of identically distributed random variables. Suppose that 
is a slowly varying function at
, and also assume that for each
, the function 
is bounded on the interval
. Suppose 
and
; and if 
then suppose also that
. Then
(2.1)
and
(2.2)
are equivalent.
For 
we also have the following theorem under adding the condition that 
is a monotone nondecreasing function.
Theorem 2.2. Let 
be a 
-mixing sequence of identically distributed random variables. Let 
is a slowly varying function at 
and monotone non-decreasing function. Suppose
; and if 
then suppose also that
. Then
(2.3)
and
(2.4)
are equivalent.
Taking 
and 
respectively in Theorems 2.1 and 2.2 we can immediately obtain the following corollaries.
Corollary 2.1. Let 
be a 
-mixing sequence of identically distributed random variables. Suppose 
and
; and if 
then suppose also that
. Then
and
are equivalent.
Corollary 2.2. Let 
be a 
-mixing sequence of identically distributed random variables. Suppose 
and
; and if 
then suppose also that
. Then
and
are equivalent.
Remark 2.1. When 
i.i.d., Corollary 2.5 becomes the Baum and Katz [12] complete convergence theorem. So Theorems 2.1 and 2.2 extend and improve the Baum and Katz complete convergence theorem from the i.i.d. case to 
-mixing sequences.
Remark 2.2. Letting 
take various forms in Theorems 2.1 and 2.2, we can get a variety of pairs of equivalent statements, one involving a moment condition and the other involving a complete convergence condition.
Proof of Theorem 2.1.
. Let
,
. Firstly, we prove that
(2.5)
By Lemma 2.1 and (2.1), it is easy to show that
(2.6)
i) For
, we have
, and
.
Let 
in (2.6), by
,
ii) For
, let 
in (2.6), then
and
. Hence
iii) For
,
Noting
, let 
in (2.6). By
and
, we get
By 
and the Kronecker lemma,
Hence (2.5) holds. So to prove (2.2) it suffices to prove that
(2.7)
and
,
(2.8)
By Lemmas 2.1 (i), (iii), (2.1), and for each
, the function 
is bounded on the interval
,
i.e., (2.7) holds.
By the Markov inequality, Lemma 1.2, Lemmas 2.1 (i), (iv), (2.1), and for each
, the function 
is bounded on the interval
,
Hence, (2.8) holds.
Now we prove that (2.2) 
(2.1). Obviously (2.2) implies
(2.9)
Noting
, by Lemma 2.1 (ii), we have
Thus,
Therefore, for sufficiently large n,
which, in conjunction with Lemma 1.2, gives
Putting this one into (2.9), we get furthermore
Thus, by Lemmas 2.1 (i), (iii),
This completes the proof of Theorem 2.1.
Proof of Theorem 2.2. (2.3) 
(2.4). Let
, the method of proof of Theorem 2.2 is similar to method used to prove the above Theorem 2.1. Only the method of prove of (2.5) is not the same. In what follows, we prove that (2.5) holds. Since 
is a monotone non-decreasing function, we have
Hence, by (2.3)，
(2.10)
i) For
, by 
and (2.10),
ii) For
, i.e., 
,
from the Kronecker lemma and
Hence (2.5) holds. The rest of the proof is similar to the corresponding part of the proof of Theorem 2.1, so we omit it.
3. Weak Law of Large Numbers
Theorem 3.1. Suppose
. Let 
be a 
-mixing sequence of identically distributed random variables satisfying
(3.1)
Then
(3.2)
Remark 3.1. When 
and 
i.i.d., then Theorem 3.1 is the weak law of large numbers (WLLN) due to Feller [11]. So, Theorem 3.1 extends the sufficient part of the Feller’s WLLN from the i.i.d. case to a 
-mixing setting.
Proof of Theorem 3.1. Let 
for 
and
. Then, for each
,
are 
-mixing identically distributed random variables and for every
,
via (3.1). So that (3.1) entails
Thus, to prove (3.2) it suffices to verify that
(3.3)
By (3.1) and the Toeplitz lemma,
Thus, together with 
for
, we have
which, in conjunction with Lemma 1.1, yields for every
,
Thus
i.e. (3.3) holds.
4. Examples
In this section, we give two examples to show our Theorems.
Example 4.1. Let 
be a 
-mixing sequence of identically distributed random variables. Suppose 
and
; and if 
then suppose also that
. Assume that 
and 
has a distribution with
.
Is easy to verify that 
satisfies the conditions of Theorems 2.1 and 2.2, and
.
Thus, by Theorems 2.1 and 2.2,
.
Example 4.2. Suppose
. Let 
be a 
-mixing sequence of identically distributed random variables. Assume that 
has a distribution with
then obviously,
Thus, by Theorem 3.1,
5. Acknowledgements
The work is supported by the National Natural Science Foundation of China (11061012), project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ([2011] 47), the Guangxi China Science Foundation (2012GXNSFAA053010), and the support program of Key Laboratory of Spatial Information and Geomatics (1103108-08).