ABSTRACT

In this paper, the complete convergence and weak law of large numbers are established for -mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to -mixing sequences of random variables without necessarily adding any extra conditions.

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).

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