A supplement to the convergence rate in a theorem of Heyde

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.

Let be a sequence of i.i.d. random variables, set , and . Heyde [1] proved that

whenever and .

There are various extensions of this result: Chen [2], Gut and Spǎtara [3], Lanzinger and Stadtmüller [4]. Liu and Lin [5] introduced a new kind of complete moment convergence; Klesov [6] studied the rate of approximation of by and proved the following Theorem A.

Theorem ALetbe a sequence of i.i.d. random variables with zero mean, if, and, then

Recently, He and Xie [7] obtained Theorem B which improved Theorem A. Gut and Steinebach [8] extended the results of Klesov [6].

Theorem BLetbe a sequence of i.i.d. random variables, and, if

then

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.

Theorem 1.1Letbe a sequence of i.i.d. random variables with zero mean and, iffor some function, and

then

where, .

Theorem 1.2Under the conditions of Theorem 1.1, and, then

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.

Before we prove the main results we state some lemmas. Lemma 2.1 is from [7]. is the standard normal distribution function, .

Lemma 2.1Letbe a sequence of i.i.d. standard normal distribution random variables. Then

Ifis a sequence of independent random variables with zero mean and finite variance, and put, , Bikelis[9]obtained 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.

Lemma 2.2Letbe a sequence of i.i.d. random variables with zero mean and. Then for any given, we have

whereis the distribution function of a random variableX.

Proof of Theorem 1.1 Without loss of generality, we suppose that , , and write

where

Applying Lemma 2.1, we obtain

then

here . By Lemma 2.2,

where

We obtain

Firstly, we estimate . Note that

Applying the condition , we have

Therefore, for any , there is an integer such that , whenever . Hence

where . For , noting that , we have the following inequality:

Next, we estimate the second term of (2.2). Note that

For , we can write

Noting that is nondecreasing in the interval , we have

where .

Similarly, we can obtain

where .

For , we write

Using the properties of by simple calculation, it follows that

and

From (2.2) to (2.8), we conclude that

Since

and

by (2.9), we have

This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2 By the conditions , and , for any , there is an integer such that , whenever . We have

and

By (2.9)-(2.11), note that , as , we have

This completes the proof of Theorem 1.2. □

Remark 2.1 If , , then , . By Theorem 1.2, we get

Remark 2.2 If , , then , , , . By (2.9), we get

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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