Let be a sequence of i.i.d. random variables with zero mean, set , , and . In this paper, the authors discuss the rate of approximation of by under suitable conditions, improve the results of Klesov (Theory Probab. Math. Stat. 49:83-87, 1994), and extend the work He and Xie (Acta Math. Appl. Sin. 2012, doi:10.1007/s10255-012-0138-6).
MSC: 60F15, 60G50.
Keywords:convergence rate; i.i.d. random variable; theorem of Heyde
1 Introduction and main results
Let be a sequence of i.i.d. random variables, set , and . Heyde  proved that
There are various extensions of this result: Chen , Gut and Spǎtara , Lanzinger and Stadtmüller . Liu and Lin  introduced a new kind of complete moment convergence; Klesov  studied the rate of approximation of by and proved the following Theorem A.
Let G be the set of functions that are defined for all real x and satisfy the following conditions: (a) is nonnegative, even, nondecreasing in the interval , and for ; (b) is nondecreasing in the interval .
Let be the set of functions satisfying the supplementary condition (c) . Obviously, the function with belongs to and does not belong to if . The purpose of this paper is to generalize Theorem B to the case where the condition is replaced by a more general condition in which the function g belongs to some subset of G. Denote , is a nonnegative nonincreasing function in the interval , and with . Now we state our results as follows.
2 Proofs of the main results
Before we prove the main results we state some lemmas. Lemma 2.1 is from . is the standard normal distribution function, .
Ifis a sequence of independent random variables with zero mean and finite variance, and put, , Bikelisobtained the following inequality:
for everyx, whereis the distribution function of the random variable. By applying the above inequality to the sequence of i.i.d. random variables with zero mean and variance 1, and letting, we have the following lemma.
Applying Lemma 2.1, we obtain
Next, we estimate the second term of (2.2). Note that
Similarly, we can obtain
From (2.2) to (2.8), we conclude that
by (2.9), we have
This completes the proof of Theorem 1.1. □
This completes the proof of Theorem 1.2. □
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
The authors are very grateful to the referees and editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper.
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