On Strong Law of Large Numbers for Dependent Random Variables

Throughout this paper, let denote the set of nonnegative integer, let be a sequence of random variables defined on probability space ( ), and put . The symbol will denote a generic constant ( ) which is not necessarily the same one in each appearance.

In [1], Jajte studied a large class of summability method as follows: a sequence is summable to by the method ( ) if

(11)

The main result of Jajte is as follows.

Theorem 1.1.

Let be a positive, increasing function and a positive function such that satisfies the following conditions.

(1)For some , is strictly increasing on with range .

(2)There exist C and a positive integer such that , .

(3)There exist constants and such that , .

Then, for i.i.d. random variables ,

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where is the inverse of function , is the indicator of event .

Motivated by Jajte [1], the present paper is devoted to the study of the limiting behavior of sums when are dependent RVs In particular, we willl consider the case when are NA RVs and obtain some general results on the complete convergence of dependent RVs First, we shall give some definitions.

Definition 1.2.

A finite family of random variables is said to be negatively associated (abbreviated NA) if, for every pair of disjoint subsets and of , we have

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whenever and are coordinatewise increasing and the covariance exists.

Definition 1.3.

A finite family of random variables is said to be positively associated (abbreviated PA) if

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whenever and are coordinatewise increasing and the covariance exists.

An infinite family of random variables is NA (resp., PA) if every finite subfamily is NA (resp., PA).

Let be the -algebra generated by RVs ,   .

Definition 1.4.

A sequence of random variables is said to be -dependence if and are independent for all and such that .

Definition 1.5.

A sequence of random variables is said to be -mixing (or uniformly strong mixing), if

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These concepts of dependence were introduced by Esary et al. [2] and Joag-Dev and Proschan [3]. Their basic properties may be found in [2, 3] and the references therein.

Definition 1.6.

Let be a sequence of random variables which is said to be: uniformly bounded by a random variable (we write ) if there exists a constant , for almost every , such that

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Remark 1.7.

The uniformly bounded random variables in (1.6) can be insured by moment conditions. For example, if

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then there exists a uniformly bounded random variable such that .

The structure of this paper is as follows. Some needed technical results will be presented in Section 2. The strong law of large numbers for NA RVs will be established in Section 3. The Spitzer and Hus-Robbins-type law of large numbers will be presented in Sections 4 and 5, respectively.

We now present some terminologies and lemmas. The following six properties are listed for reference in obtaining the main results in the next sections. Detailed proofs can be founded in the cited references.

Lemma 2.1 (cf. [4]) (three-series theorem for NA).

Let   be NA. Let and let . In order that converges a.s., it is sufficient that

(1) ,

(2) converges,

(3) .

Lemma 2.2 (cf. [4]).

Let be NA with , , then for , for all

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Lemma 2.3 (cf. [5]).

Let be -dependence with , , then

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Lemma 2.4 (cf. [5]).

Let be -mixing with , , then

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Lemma 2.5 (cf. [6]).

Let be PA with , , then

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Furthermore, if

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then

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where .

Lemma 2.6.

Let be a sequence of random variables and a random variable. If , then for all ,

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Proof.

By the integral equality

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it follows that

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Lemma 2.7 (cf. [7]).

Let be a sequence of events defined on . If , then , if and for , then .

Theorem 3.1.

Let , and be as in Theorem 1.1, and let be a sequence of negatively associated random variables with . Assume that . If , then

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Conversely, let be a sequence of identically distributed NA random variables, if (3.1) is true, then .

Proof.

Assume that . To prove (3.1) by applying the Kronecker lemma, it suffices to show that

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Here, we shall use the three-series theorem for NA RVs.

Let . Then, by , we have

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which shows that , and

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Therefore, from , it follows that

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To this end we estimate the series

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Conversely,since are identically NA RVs. If (3.1) holds, that is,

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It follows that

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which shows that . Hence,

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where and .

Since is still an NA sequence. Defining the following events,

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we have , and , for . By Lemma 2.7, if a.s., then and . Therefore,

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which is equivalent to .

These complete the proof of Theorem 3.1.

Theorem 3.1 also includes a particular case of logarithmic means, we can establish the following.

Corollary 3.2.

Let be a sequence of NA RVs with and . If , then, one has

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Proof.

Let , , that is, . In this case, as , therefore , for .

Corollary 3.3.

Let be a sequence of NA RVs with and . If , then, for every , , one has

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Remark 3.4.

As pointed out by Jajte [1], Theorem 3.1 includes several regular summability methods such as (1) the Kolmogorov SLLN ; (2) the classical MZ SLLN .

Since the definition of complete convergence was introduced by Hsu and Robins, there have been many authors who devote themselves to the study of the complete convergence for sums of independent and dependent RVs and obtain a series of elegant results, see [4, 8] and reference therein.

We say that the Hsu-Robbins [9] law of large numbers (LLN) is valid if, for all ,

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and the Spitzer [10] LLN is valid if, for all ,

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Theorem 4.1.

Let be as in Theorem 1.1, and let be a sequence of NA random variables with . Assume that . If , then for all ,

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Conversely, let be a sequence of identically distributed NA random variables, if (4.3) is true, then and .

Proof.

For , let , , then for every

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Note that

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For the first term on the RHS of (4.5), by Markov inequality and Lemma 2.2 and (3.6), we have

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For the second term on the RHS of (4.5), since

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hence,

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For the third term on the RHS of (4.5), we have, by (3.5),

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Therefore, (4.3) follows.

Conversely, since imply that ., hence from Theorem 3.1, we have . These complete the proof of Theorem 4.1.

Corollary 4.2.

Under the assumptions of Theorem 4.1, one has

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Proof.

Denote , and noticing that

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hence, similarly to the proof of Theorem 4.1, we obtain (4.10).

Analogously, we can prove the following corollaries, and omit the details.

Corollary 4.3.

Let be a sequence of -mixing random variables with . Assume that . If , and

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then (4.3) holds.

Corollary 4.4.

Let be a sequence of -dependent random variables with . Assume that . If , then (4.3) holds.

Corollary 4.5.

Let be a sequence of PA random variables with . Assume that . If , and

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then (4.3) holds.

Theorem 5.1.

Let , be define as in Theorem 1.1, but the following condition (3) is replaced by

(3′) There exist constants such that , , , and let be a sequence of NA random variables with . Assume that . If , then for all ,

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Proof.

From the previous section, we know that to prove Theorem 5.1, we need only to prove the convergence of the following three series.

First, note that , we have

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Hence, as . Next, by (4.4), we have

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Last, from the definition of and the NA's property, we know that remains a sequence of NA RVs. By applying Lemma 2.2 and inequality, we have

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It is easy to see that

(55)

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These complete the proof of Theorem 5.1.