Complete convergence and complete moment convergence for a class of random variables
© Wang and Hu; licensee Springer 2012
Received: 14 July 2012
Accepted: 1 October 2012
Published: 12 October 2012
In this paper, we establish the complete convergence and complete moment convergence and obtain the equivalence of the complete convergence and complete moment convergence for the class of random variables satisfying a Rosenthal-type maximal inequality. Baum-Katz-type theorem and Hsu-Robbins-type theorem are extended to the case of this class of random variables.
The concept of complete convergence was introduced by Hsu and Robbins  as follows. A sequence of random variables is said to converge completely to a constant C if for all . In view of the Borel-Cantelli lemma, this implies that almost surely (a.s.). The converse is true if the are independent. Hsu and Robbins  proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Erdös  proved the converse. The result of Hsu-Robbin-Erdös is a fundamental theorem in probability theory which has been generalized and extended in several directions by many authors. One of the most important generalizations is that by Baum and Katz  for the strong law of large numbers as follows.
Theorem A (Baum and Katz )
Many authors studied the Baum-Katz-type theorem for dependent random variables; see, for example, Peligrad  for a strong stationary ρ-mixing sequence, Peligrad and Gut  for a -mixing sequence, Stoica [6, 7] for a martingale difference sequence, Stoica  for bounded subsequences, Wang and Hu  for φ-mixing random variables, and so forth.
The inequality above has been obtained for dependent random variables by many authors. See, for example, Shao  for negatively associated random variables, Utev and Peligrad  for -mixing random variables, Wang et al.  for φ-mixing random variables with the mixing coefficients satisfying certain conditions, and so forth.
The purpose of this work is to obtain complete convergence and complete moment convergence for a sequence of random variables satisfying a Rosenthal-type maximal inequality.
Throughout the paper, let be a sequence of random variables defined on a fixed probability space . Let be the indicator function of the set A. Denote , , , (). The symbol C denotes a positive constant which may be different in various places.
The following definition will be used frequently in the paper.
for all and .
In this paper, the present investigation is to provide the complete convergence and complete moment convergence for a sequence of the class of random variables satisfying a Rosenthal-type maximal inequality and prove the equivalence of the complete convergence and complete moment convergence. Baum-Katz-type theorem and Hsu-Robbins-type theorem are extended to the case of the class of random variables satisfying a Rosenthal-type maximal inequality. As a result, the Marcinkiewicz-Zygmund strong law of large numbers for the class of random variables is obtained.
2 Main results
Furthermore, (2.2) is equivalent to (2.3).
For , we have the following theorem.
Remark 2.1 In (2.1), is a monotone transformation of . If is a sequence of independent random variables, then (2.1) is clearly satisfied. There are many sequences of dependent random variables satisfying (2.1) for all . Examples include sequences of NA random variables (see Shao ), -mixing random variables (see Utev and Peligrad ), φ-mixing random variables with the mixing coefficients satisfying certain conditions (see Wang et al. ), -mixing random variables with the mixing coefficients satisfying certain conditions (see Wang and Lu ).
Remark 2.2 In Theorem 2.1, we not only generalize the Baum-Katz-type theorem for the class of random variables satisfying (2.1), but also consider the case . Furthermore, if we take and , then we can get the Hsu-Robbins-type theorem (see Hsu and Robbins ) for the class of random variables satisfying (2.1).
In this section, the following lemmas are very useful to prove the main results of the paper.
Lemma 3.1 (cf. Wu )
where and are positive constants.
We will consider the following three cases:
Case 1. , and .
Case 2. , and .
Case 3. , and .
Take . Note that if . Similar to the proofs of (3.5), it follows that . From the statements above, (3.1) is proved. The proof of the lemma is completed. □
Lemma 3.3 (cf. Sung )
4 The proofs of main results
We consider the following three cases:
Case 1. , and .
Take , which implies that .
Case 2. , and .
Case 3. , and .
Take . Note that if . Similar to the proof of (4.4), it follows that . From all the statements above, we have proved (2.2).
Hence (2.3) holds.
Hence, by (4.5), (2.3) implies (2.2). The proof of the theorem is completed. □
The rest of the proof is similar to that of Theorem 3.1 in Dung and Tien  and is omitted. □
Hence, (2.4) holds. □
The desired result (2.5) follows from (2.2) and (4.6) immediately. □
The authors are most grateful to the editor Andrei Volodin and anonymous referees for careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper. The research was supported by the National Natural Science Foundation of China (11171001, 11201001, 11126176), Natural Science Foundation of Anhui Province (1208085QA03), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Academic Innovation Team of Anhui University (KJTD001B), Doctoral Research Start-up Funds Projects of Anhui University, and the Talents Youth Fund of Anhui Province Universities (2011SQRL012ZD).
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