Complete Convergence and Weak Law of Large Numbers for  Ρ -mixing Sequences of Random Variables

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.


Introduction
where (for a given positive integer ) this sup is taken over all pairs of nonempty finite subsets such that dist .n .Without loss of generality we may assume that n is such that (see [1]).Here we give two examples of the practical application of Example 1.1.According to the proof of Theorem 2 in [2] and Remark 3 in [1], if spectral density has the property that .Therefore, instantaneous functions   , 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][5][6][7][8][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: and and , -mixing sequence of random variables.Then for any , there exists a positive constant c such that for all , . Without loss of generality, assume that n  .By the Cauchy-Schwarz inequality and Lemma 1.2, , and n n n denote that there exists a constant c such that , and is said to be a slowly varying function at if for any , there exist positive constants and 2 (depending only on  , and the ) such that for any positive number k, , there exist positive constants and 2 (depending only on  , and the ) such that for any positive number k, quence of identically distributed random variables.Suppose that   0 l x  is a slowly varying function at  , and also assume that for each ; and if   then suppose also that and we also have the following theorem under adding the condition that is a monotone nondecreasing function.
Theorem 2.2.Let n be a  ; ; and if Taking and respectively in Theorems 2.1 and 2.2 we can immediately obtain the following corollaries.

Noting
  , by Lemma 2.1 (ii), we have Therefore, for sufficiently large n, Putting this one into (2.9),we get furthermore , 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 and (2.10), 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.

Weak Law of Large Numbers
-mixing identically distributed random variables and for every  , Thus, together with which, in conjunction with Lemma 1.1, yields for every

Examples
In this section, we give two examples to show our Theorems.


Is easy to verify that  l x satisfies the conditions of Theorems 2.1 and 2.2, and    f t , then the sequence Let be a probability space.The random variables we deal with are all defined on .Let n be a sequence of random variables.For each nonempty and except in the trivial case where all of the random variables

Remark 2 . 1 .
When  n 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   l xand the other involving a complete convergence condition.Proof of Theorem 2.1.
When and n 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

Example 4 . 1 .
Let n be a ;X n    -mixing sequence of identically distributed random variables.Sup-

1 -
mixing sequence of identically distributed random variables.Assume that X has a distribution with