Convergence Theorems for Partial Sums of Arbitrary Stochastic Sequences
© Xiaosheng Wang and Haiying Guo. 2010
Received: 27 May 2010
Accepted: 20 October 2010
Published: 24 October 2010
By using Doob's martingale convergence theorem, this paper presents a class of strong limit theorems for arbitrary stochastic sequence. Chow's two strong limit theorems for martingale-difference sequence and Loève's and Petrov's strong limit theorems for independent random variables are the particular cases of the main results.
Almost sure behavior of partial sums of random variables has enjoyed both a rich classical period and a resurgence of research activity. Some famous researchers, such as Borel, Kolmogorov, Khintchine, Loève, Chung, and so on, were interested in convergence theorem of partial sums of random variables and obtained lots of classical results for sequences of independent random variables and martingale differences. For a detailed survey of strong limit theorems of sequences for random variables, interested readers can refer to the books [1, 2].
In recent years, some work has been done on the strong limit theorems for arbitrary stochastic sequences. Liu and Yang  established two strong limit theorems for arbitrary stochastic sequences, which generalized Chung's  strong law of large numbers for sequence of independent random variables as well as Chow's  strong law of large numbers for sequence of martingale differences. Then, Yang  established two more general strong limit theorems in 2007, which generalized a result by Jardas et al.  for sequences of independent random variables and the results by Liu and Yang  for arbitrary stochastic sequences in 2003. In 2008, W. Yang and X. Yang  proved two strong limit theorems for stochastic sequences, which generalized results by Freedman , Isaac,  and Petrov . Qiu and Yang  established another type strong limit theorem for stochastic sequence in 1999. Then, Wang and Guo  extended the main result of Qiu and Yang in 2009. In addition, Wang and Yang  established a strong limit theorem for arbitrary stochastic sequences in 2005, which generalized Chow's  series convergence theorem for sequence of martingale differences. Then, Qiu  extended the result of Wang and Yang in 2008.
The purpose of this paper is to discuss further the strong limit theorems for arbitrary stochastic sequences. By using Doob's  convergence theorem for martingale-difference sequence, we establish a class of new strong limit theorems for stochastic sequences. Chow's two strong limit theorems for martingale-difference sequence, Loève's series convergence theorem, and Petrov's strong law of large numbers for sequences of independent random variables are the particular cases of this paper. In addition, the main theorems of this paper extend the main results by Wang and Guo in 2009, Qiu and Yang in 1999, and the result by Wang and Yang in 2005, respectively. The remainder of this paper is organized as follows. In Section 2, we present the main theorems of this paper. In Section 3, the proofs of the main theorems in this paper are presented.
2. Main Theorems
In this section, we will introduce the main results of this paper.
Corollary 2.2 (Chow).
Corollary 2.4 (Loève).
Corollary 2.5 (Petrov).
Corollary 2.9 (Chow).
where the log is to the base 2.
3. Proofs of Theorems
We first give a lemma.
Lemma 3.1 (see ).
Proof of Theorem 2.1.
The following argument breaks down into two cases.
It follows from (3.12) and (3.14) that (2.5) holds.
It follows from (3.12) and (3.16) that (2.6) holds. The theorem is proved.
Proof of Theorem 2.3.
Proof of Theorem 2.6.
Proof of Theorem 2.8.
This work was supported by National Natural Science Foundation of China no. 11071104 and Hebei Natural Science Foundation no. F2010001044.
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