On Strong Law of Large Numbers for Dependent Random Variables
© Zhongzhi Wang. 2011
Received: 16 December 2010
Accepted: 3 March 2011
Published: 14 March 2011
We discuss strong law of large numbers and complete convergence for sums of uniformly bounded negatively associate (NA) random variables (RVs). We extend and generalize some recent results. As corollaries, we investigate limit behavior of some other dependent random sequence.
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.
The main result of Jajte is as follows.
Motivated by Jajte , 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.
An infinite family of random variables is NA (resp., PA) if every finite subfamily is NA (resp., PA).
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. ) (three-series theorem for NA).
Lemma 2.2 (cf. ).
Lemma 2.3 (cf. ).
Lemma 2.4 (cf. ).
Lemma 2.5 (cf. ).
Lemma 2.7 (cf. ).
3. Strong Law of Large Numbers
Here, we shall use the three-series theorem for NA RVs.
These complete the proof of Theorem 3.1.
Theorem 3.1 also includes a particular case of logarithmic means, we can establish the following.
As pointed out by Jajte , Theorem 3.1 includes several regular summability methods such as (1) the Kolmogorov SLLN ; (2) the classical MZ SLLN .
4. Spitzer Type Law of Large Numbers
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.
Therefore, (4.3) follows.
hence, similarly to the proof of Theorem 4.1, we obtain (4.10).
Analogously, we can prove the following corollaries, and omit the details.
then (4.3) holds.
then (4.3) holds.
5. Hsu-Robbins Type Law of Large Numbers
From the previous section, we know that to prove Theorem 5.1, we need only to prove the convergence of the following three series.
These complete the proof of Theorem 5.1.
This work is supported by the National Nature Science Foundation of China (no. 11071104) and the Anhui high Education Research Grant (no. KJ2010A337). The author expresses his sincere gratitude to the referees and the editors for their hospitality.
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