Abstract
We investigate the complete moment convergence for maximal partial sum of arrays of rowwise -mixing random variables under some more general conditions. The results obtained in the paper generalize and improve some known ones.
1. Introduction
Let be a sequence of random variables defined on a fixed probability space . Let and be positive integers. Write . Given -algebras in , let Define the -mixing coefficients by
A random variable sequence is said to be -mixing if as . is called mixing coefficient. A triangular array of random variables is said to be an array of rowwise -mixing random variables if, for every , is a -mixing sequence of random variables. The notion of -mixing random variables was introduced by Dobrushin [1] and many applications have been found. See, for example, Utev [2] for central limit theorem, Gan and Chen [3] for limit theorem, Peligrad [4] for weak invariance principle, Shao [5] for almost sure invariance principles, Chen and Wang [6], Shen et al. [7, 8], Wu [9], and Wang et al. [10] for complete convergence, Hu and Wang [11] for large deviations, and so forth. When these are compared with corresponding results of independent random variable sequences, there still remains much to be desired.
Definition 1. A sequence of random variables is said to converge completely to a constant if, for any , In this case, one writes completely. This notion was given first by Hsu and Robbins [12].
Definition 2. Let be a sequence of random variables and , , and . If then the above result was called the complete moment convergence by Chow [13].
Let be an array of rowwise -mixing random variables with mixing coefficients in each row, let be a sequence of positive real numbers such that , and let be a sequence of positive even functions such that for some and each . In order to prove our results, we mention the following conditions: where is a positive integer.
The following are examples of function satisfying assumption (5): for some or for . Note that these functions are nonmonotone on , while it is simple to show that, under condition (5), the function is an increasing function for . In fact, , , and as ; then we have .
Recently Gan et al. [14] obtained the following complete convergence for -mixing random variables.
Theorem A. Let be a sequence of -mixing mean zero random variables with , let be a sequence of positive real numbers with , and let be a sequence of nonnegative even functions such that as and and as , where . If the following conditions are satisfied: where , , then
For more details about this type of complete convergence, one can refer to Gan and Chen [3], Wu et al. [15], Wu [16], Huang et al. [17], Shen [18], Shen et al. [19, 20], and so on. The purpose of this paper is extending Theorem A to the complete moment convergence, which is a more general version of the complete convergence, and making some improvements such that the conditions are more general. In this work, the symbol always stands for a generic positive constant, which may vary from one place to another.
2. Preliminary Lemmas
In this section, we give the following lemma which will be used to prove our main results.
Lemma 3 (cf. Wang et al. [10]). Let be a sequence of -mixing random variables satisfying , . Assume that , and , for each . Then there exists a constant depending only on and such that for every and . In particular, one has for every .
3. Main Results and Their Proofs
Let be an array of rowwise -mixing random variables and let be the mixing coefficient of for any . Our main results are as follows.
Theorem 4. Let be an array of rowwise -mixing random variables satisfying and let be a sequence of positive real numbers such that . Also, let be a positive even function satisfying (5) for . Then under conditions (6) and (7), one has
Proof. Firstly, let us prove the following statements from conditions (5) and (7).
(i) For , ,
(ii) For ,
For , denote . It is easy to check that
To prove (14), it suffices to prove that and . Now let us prove them step by step. Firstly, we prove that .
For all , define
then for all , it is easy to have
By (5), (6), (7), and (15) we have
From (19) and (20), it follows that, for large enough,
Hence we only need to prove that
For , it follows by (15) that
For , take . Since , , we have by Markov inequality, Lemma 3, -inequality, and (16) that
Next we prove that . Denote , , and . Obviously,
Hence,
For , by (15), we have
Now let us prove that . Firstly, it follows by (6) and (15) that
Therefore, for sufficiently large,
Then for sufficiently large,
Let . By (30), Lemma 3, and -inequality, we can see that
For , since , we have
Since , by (16), it implies . Now we prove that . Since and as , by (15) we have
Let in . Note that, for ,
Then by (15) and , we have
This completes the proof of Theorem 4.
Theorem 5. Let be an array of rowwise -mixing random variables satisfying and let be a sequence of positive real numbers such that . Also, let be a positive even function satisfying (5) for and . Then conditions (6)–(8) imply (14).
Proof. Following the notation, by a similar argument as in the proof of Theorem 4, we can easily prove that , and that (19) and (20) hold. To complete the proof, we only need to prove that .
Let and . By (30), Markov inequality, Lemma 3, and the -inequality we can get
For , we have
By a similar argument as in the proof of and (replacing the exponent by ), we can get and .
For , since , we can see that
Since , from (8) we have
Next we prove that . To start with, we consider the case . Since , by (15), we have
Finally, we prove that in the case . Since and , we have by (15) that
Thus we get the desired result immediately. The proof is completed.
Corollary 6. Let be an array of rowwise -mixing mean zero random variables with , . If, for some and , where , , then, for any ,
Proof. Put , , , and .
Since , , then
It follows by (42) and that
Since , by Jensen's inequality it follows that
Clearly . Take such that . Therefore,
Combining Theorem 5 and (45)–(47), we can prove Corollary 6 immediately.
Remark 7. Noting that in this paper we consider the case , which has a more wide scope than the case , in Gan et al. [14]. In addition, compared with -mixing random variables, the arrays of -mixing random variables not only have many related properties, but also have a wide range of application. So it is very significant to study it.
Remark 8. Under the condition of Theorem 4, we have Then we can obtain (11) directly. In this case, condition (10) is not needed. Especially, for , the conditions of Theorem 4 are weaker than Theorem A. So Theorem 4 generalizes and improves it.
Remark 9. Note that Theorem A only considers , while Theorem 5 considers . In addition, (14) implies (11), so Theorem 5 generalizes the corresponding result of Theorem A.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are most grateful to the Editor Ming Mei and anonymous reviewer for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by National Natural Science Foundation of China (11201001, 11126176) and the Students Innovative Training Project of Anhui University (201310357004).