A supplement to the convergence rate in a theorem of Heyde
© He and Xie; licensee Springer. 2012
Received: 27 May 2012
Accepted: 20 August 2012
Published: 4 September 2012
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).
1 Introduction and main results
whenever and .
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.
Throughout this paper, we suppose that C denotes a constant which only depends on some given numbers and may be different at each appearance, and that denotes the integer part of x.
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, .
for every x, 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.
whereis the distribution function of a random variable X.
This completes the proof of Theorem 1.1. □
This completes the proof of Theorem 1.2. □
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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