Open Journal of Statistics
Vol.2 No.5(2012), Article ID:25544,7 pages DOI:10.4236/ojs.2012.25062
Complete Convergence and Weak Law of Large Numbers for -Mixing Sequences of Random Variables
1College of Science, Guilin University of Technology, Guilin, China
2Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin, China
Email: wqy666@glut.edu.cn
Received September 5, 2012; revised October 7, 2012; accepted October 20, 2012
Keywords: -Mixing Sequence of Random Variables; Complete Convergence; Weak Law of Large Number
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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