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A supplement to the convergence rate in a theorem of Heyde
Journal of Inequalities and Applications volume 2012, Article number: 195 (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
Let be a sequence of i.i.d. random variables, set , and . Heyde  proved that
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.
Theorem A Letbe a sequence of i.i.d. random variables with zero mean, if, and, then
Theorem B Letbe a sequence of i.i.d. random variables, and, if
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 1.1 Letbe a sequence of i.i.d. random variables with zero mean and, iffor some function, and
Theorem 1.2 Under 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.
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, .
Lemma 2.1 Letbe 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, , Bikelisobtained the following inequality:
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.
Lemma 2.2 Letbe 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 variable X.
Proof of Theorem 1.1 Without loss of generality, we suppose that , , and write
Applying Lemma 2.1, we obtain
here . By Lemma 2.2,
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
Similarly, we can obtain
For , we write
Using the properties of by simple calculation, it follows that
From (2.2) to (2.8), we conclude that
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
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
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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.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Cite this article
He, J., Xie, T. A supplement to the convergence rate in a theorem of Heyde. J Inequal Appl 2012, 195 (2012). https://doi.org/10.1186/1029-242X-2012-195
- convergence rate
- i.i.d. random variable
- theorem of Heyde