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Discrete Dynamics in Nature and Society
Volumeย 2011ย (2011), Article IDย 185160, 8 pages
http://dx.doi.org/10.1155/2011/185160
Research Article

A Note on the Inverse Moments for Nonnegative ๐œŒ-Mixing Random Variables

School of Mathematical Science, Anhui University, Hefei 230039, China

Received 24 June 2011; Accepted 15 July 2011

Academic Editor: Tetsujiย Tokihiro

Copyright ยฉ 2011 Aiting Shen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Wu et al. (2009) studied the asymptotic approximation of inverse moments for nonnegative independent random variables. Shen et al. (2011) extended the result of Wu et al. (2009) to the case of ๐œŒ-mixing random variables. In the paper, we will further study the asymptotic approximation of inverse moments for nonnegative ๐œŒ-mixing random variables, which improves the corresponding results of Wu et al. (2009), Wang et al. (2010), and Shen et al. (2011) under the case of identical distribution.

1. Introduction

Firstly, we will recall the definition of ๐œŒ-mixing random variables.

Let {๐‘‹๐‘›,๐‘›โ‰ฅ1} be a sequence of random variables defined on a fixed probability space (ฮฉ,โ„ฑ,๐‘ƒ). Let ๐‘› and ๐‘š be positive integers. Write โ„ฑ๐‘š๐‘›=๐œŽ(๐‘‹๐‘–,๐‘›โ‰ค๐‘–โ‰ค๐‘š) and โ„ฑ๐’ฎ=๐œŽ(๐‘‹๐‘–,๐‘–โˆˆ๐‘†โŠ‚โ„•). Given ๐œŽ-algebras โ„ฌ,โ„› in โ„ฑ, let ๐œŒ(โ„ฌ,โ„›)=sup๐‘‹โˆˆ๐ฟ2(โ„ฌ),๐‘Œโˆˆ๐ฟ2(โ„›)||||๐ธ๐‘‹๐‘Œโˆ’๐ธ๐‘‹๐ธ๐‘Œโˆš.Var(๐‘‹)โ‹…Var(๐‘Œ)(1.1) Define the ๐œŒ-mixing coefficients by ๐œŒ(๐‘›)=sup๐‘˜โ‰ฅ1๐œŒ๎€ทโ„ฑ๐‘˜1,โ„ฑโˆž๐‘˜+๐‘›๎€ธ,๐‘›โ‰ฅ0.(1.2)

Definition 1.1. A sequence {๐‘‹๐‘›,๐‘›โ‰ฅ1} of random variables is said to be ๐œŒ-mixing if ๐œŒ(๐‘›)โ†“0 as ๐‘›โ†’โˆž.

๐œŒ-mixing sequence was introduced by Kolmogorov and Rozanov [1]. It is easily seen that ๐œŒ-mixing sequence contains independent sequence as a special case.

The main purpose of the paper is to study the asymptotic approximation of inverse moments for nonnegative ๐œŒ-mixing random variables with identical distribution.

Let {๐‘๐‘›,๐‘›โ‰ฅ1} be a sequence of independent nonnegative random variables with finite second moments. Denote ๐‘‹๐‘›=โˆ‘๐‘›๐‘–=1๐‘๐‘–๐ต๐‘›,๐ต2๐‘›=๐‘›๎“๐‘–=1Var๐‘๐‘–.(1.3) It is interesting to show that under suitable conditions the following equivalence relation holds, namely, ๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐‘Ÿโˆผ๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธโˆ’๐‘Ÿ,๐‘›โŸถโˆž,(1.4) where ๐‘Ž>0 and ๐‘Ÿ>0 are arbitrary real numbers.

Here and below, for two positive sequences {๐‘๐‘›,๐‘›โ‰ฅ1} and {๐‘‘๐‘›,๐‘›โ‰ฅ1}, we write ๐‘๐‘›โˆผ๐‘‘๐‘› if ๐‘๐‘›๐‘‘๐‘›โˆ’1โ†’1 as ๐‘›โ†’โˆž. ๐ถ is a positive constant which can be different in various places.

The inverse moments can be applied in many practical applications. For example, they may be applied in Stein estimation and poststratification (see [2, 3]), evaluating risks of estimators and powers of tests (see [4, 5]). In addition, they also appear in the reliability (see [6]) and life testing (see [7]), insurance and financial mathematics (see [8]), complex systems (see [9]), and so on.

Under certain asymptotic-normality condition, relation (1.4) was established in Theoremโ€‰โ€‰2.1 of Garcia and Palacios [10]. But, unfortunately, that theorem is not true under the suggested assumptions, as pointed out by Kaluszka and Okolewski [11]. The latter authors established (1.4) by modifying the assumptions as follows:(i)๐‘Ÿ<3 (๐‘Ÿ<4, in the i.i.d. case);(ii)๐ธ๐‘‹๐‘›โ†’โˆž, ๐ธ๐‘3๐‘›<โˆž;(iii)(๐ฟ๐‘ condition) โˆ‘๐‘›๐‘–=1๐ธ|๐‘๐‘–โˆ’๐ธ๐‘๐‘–|๐‘/๐ต๐‘๐‘›โ†’0 (๐‘=3).

Hu et al. [12] considered weaker conditions: ๐ธ๐‘๐‘›2+๐›ฟ<โˆž, where ๐‘๐‘› satisfies ๐ฟ2+๐›ฟ condition and 0<๐›ฟโ‰ค1. Wu et al. [13] applied Bernsteinโ€™s inequality and the truncated method to greatly improve the conclusion in weaker condition on moment. Wang et al. [14] extended the result for independent random variables to the case of NOD random variables. Shi et al. [15] obtained (1.4) for ๐ต๐‘›=1. Sung [16] studied the inverse moments for a class of nonnegative random variables.

Recently, Shen et al. [17] extended the result of Wu et al. [13] to the case of ๐œŒ-mixing random variables and obtained the following result.

Theorem A. Let {๐‘๐‘›,๐‘›โ‰ฅ1} be a nonnegative ๐œŒ-mixing sequence with โˆ‘โˆž๐‘›=1๐œŒ(๐‘›)<โˆž. Suppose that (i)๐ธ๐‘2๐‘›<โˆž, for all ๐‘›โ‰ฅ1;(ii)๐ธ๐‘‹๐‘›โ†’โˆž, where ๐‘‹๐‘› is defined by (1.3);(iii)for some ๐œ‚>0,๐‘…๐‘›(๐œ‚)โˆถ=๐ต๐‘›๐‘›โˆ’2๎“๐‘–=1๐ธ๐‘2๐‘–๐ผ๎€ท๐‘๐‘–>๐œ‚๐ต๐‘›๎€ธโŸถ0,๐‘›โŸถโˆž;(1.5)(iv)for some ๐‘กโˆˆ(0,1) and any positive constants ๐‘Ž,๐‘Ÿ,๐ถ,lim๐‘›โ†’โˆž๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธ๐‘Ÿ๎ƒฏ๎€ทโ‹…expโˆ’๐ถโ‹…๐ธ๐‘‹๐‘›๎€ธ๐‘ก๐‘›๎ƒฐ=0.(1.6) Then for any ๐‘Ž>0 and ๐‘Ÿ>0, (1.4) holds.

In this paper, we will further study the asymptotic approximation of inverse moments for nonnegative ๐œŒ-mixing random variables with identical distribution. We will show that (1.4) holds under very mild conditions and the condition (iv) in Theorem A can be deleted. In place of the Bernstein type inequality used by Shen et al. [17], we make the use of Rosenthal type inequality of ๐œŒ-mixing random variables. Our main results are as follows.

Theorem 1.2. Let {๐‘๐‘›,๐‘›โ‰ฅ1} be a sequence of nonnegative ๐œŒ-mixing random variables with identical distribution and let {๐ต๐‘›,๐‘›โ‰ฅ1} be a sequence of positive constants. Let ๐‘Ž>0 and ๐›ผ>0 be real numbers. ๐‘>max{2,2๐›ผ,๐›ผ+1}. Assume that โˆ‘โˆž๐‘›=1๐œŒ2/๐‘(2๐‘›)<โˆž. Suppose that (i)0<๐ธ๐‘๐‘›<โˆž, for all ๐‘›โ‰ฅ1;(ii)๐œ‡๐‘›โ‰๐ธ๐‘‹๐‘›โ†’โˆž as ๐‘›โ†’โˆž, where ๐‘‹๐‘›=๐ต๐‘›โˆ’1โˆ‘๐‘›๐‘˜=1๐‘๐‘˜;(iii)for all 0<๐œ€<1, there exist ๐‘>0 and ๐‘›0>0 such that ๐ธ๐‘1๐ผ๎€ท๐‘1>๐‘๐ต๐‘›๎€ธโ‰ค๐œ€๐ธ๐‘1,๐‘›โ‰ฅ๐‘›0.(1.7) Then (1.4) holds.

Corollary 1.3. Let {๐‘๐‘›,๐‘›โ‰ฅ1} be a sequence of nonnegative ๐œŒ-mixing random variables with identical distribution and 0<๐ธ๐‘1<โˆž. Let {๐ต๐‘›,๐‘›โ‰ฅ1} be a sequence of positive constants satisfying ๐ต๐‘›=๐‘‚(๐‘›๐›ฟ) for some 0<๐›ฟ<1 and ๐ต๐‘›โ†’โˆž as ๐‘›โ†’โˆž. Let ๐‘Ž>0 and ๐›ผ>0 be real numbers. ๐‘>max{2,2๐›ผ,๐›ผ+1}. Assume that โˆ‘โˆž๐‘›=1๐œŒ2/๐‘(2๐‘›)<โˆž. Then (1.4) holds.

By Theorem 1.2, we can get the following convergence rate of relative error in the relation (1.4).

Theorem 1.4. Assume that conditions of Theorem 1.2 are satisfied and 0<๐ธ๐‘2๐‘›<โˆž. ๐‘>max{2,4(๐›ผ+1),2๐›ผ+3}. If ๐ต๐‘›โ‰ฅ๐ถ๐‘›1/2 for all ๐‘› large enough, where ๐ถ is a positive constant, then ||๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธ๐›ผ๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผ||๎‚€๎€ทโˆ’1=๐‘‚๐‘Ž+๐ธ๐‘‹๐‘›๎€ธโˆ’1๎‚.(1.8)

Theorem 1.5. Assume that conditions of Theorem 1.2 are satisfied and 0<๐ธ๐‘2๐‘›<โˆž. ๐‘>max{2,4(๐›ผ+1),2๐›ผ+3}. Then ||๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธ๐›ผ๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผ||๎€ท๐‘›โˆ’1=๐‘‚โˆ’1/2๎€ธ.(1.9)

Taking ๐ต๐‘›โ‰ก1 in Theorem 1.2, we have the following asymptotic approximation of inverse moments for the partial sums of nonnegative ๐œŒ-mixing random variables with identical distribution.

Theorem 1.6. Let {๐‘๐‘›,๐‘›โ‰ฅ1} be a sequence of nonnegative ๐œŒ-mixing random variables with identical distribution. Let ๐‘Ž>0 and ๐›ผ>0 be real numbers. ๐‘>max{2,2๐›ผ,๐›ผ+1}. Assume that โˆ‘โˆž๐‘›=1๐œŒ2/๐‘(2๐‘›)<โˆž. Suppose that (i)0<๐ธ๐‘๐‘›<โˆž, โˆ€๐‘›โ‰ฅ1;(ii)๐œˆ๐‘›โ‰๐ธ๐‘Œ๐‘›โ†’โˆž as ๐‘›โ†’โˆž, where ๐‘Œ๐‘›=โˆ‘๐‘›๐‘˜=1๐‘๐‘˜;(iii)for all 0<๐œ€<1, there exist ๐‘>0 and ๐‘›0>0 such that ๐ธ๐‘1๐ผ๎€ท๐‘1๎€ธ>๐‘โ‰ค๐œ€๐ธ๐‘1,๐‘›โ‰ฅ๐‘›0.(1.10) Then ๐ธ(๐‘Ž+๐‘Œ๐‘›)โˆ’๐›ผโˆผ(๐‘Ž+๐ธ๐‘Œ๐‘›)โˆ’๐›ผ.

Remark 1.7. Theorem 1.2 in this paper improves the corresponding results of Wu et al. [13], Wang et al. [14], and Shen et al. [17]. Firstly, Theorem 1.4 in this paper is based on the condition ๐ธ๐‘๐‘›<โˆž, for all ๐‘›โ‰ฅ1, which is weaker than the condition ๐ธ๐‘2๐‘›<โˆž, for all ๐‘›โ‰ฅ1 in the above cited references. Secondly, {๐ต๐‘›,๐‘›โ‰ฅ1} is an arbitrary sequence of positive constants in Theorem 1.2, while ๐ต2๐‘›=โˆ‘๐‘›๐‘–=1Var๐‘๐‘– in the above cited references. Thirdly, the condition (iv) in Theorem A is not needed in Theorem 1.2. Finally, (1.7) is weaker than (1.5) under the case of identical distribution. Actually, by the condition (1.5), we can see that ๐ต๐‘›๐‘›โˆ’1๎“๐‘–=1๐ธ๐‘๐‘–๐ผ๎€ท๐‘๐‘–>๐œ‚๐ต๐‘›๎€ธโ‰ค๐œ‚โˆ’1๐ต๐‘›๐‘›โˆ’2๎“๐‘–=1๐ธ๐‘2๐‘–๐ผ๎€ท๐‘๐‘–>๐œ‚๐ต๐‘›๎€ธโŸถ0,๐‘›โŸถโˆž,(1.11) which implies that for all 0<๐œ€<1, there exists a positive integer ๐‘›0 such that ๐ต๐‘›๐‘›โˆ’1๎“๐‘–=1๐ธ๐‘๐‘–๐ผ๎€ท๐‘๐‘–>๐œ‚๐ต๐‘›๎€ธโ‰ค๐œ€๐œ‡๐‘›=๐œ€๐ต๐‘›๐‘›โˆ’1๎“๐‘–=1๐ธ๐‘๐‘–,๐‘›โ‰ฅ๐‘›0,(1.12) that is, (1.7) holds.

2. Proof of the Main Results

In order to prove the main results of the paper, we need the following important moment inequality for ๐œŒ-mixing random variables.

Lemma 2.1 (c.f. Shao [18, Corollaryโ€‰โ€‰1.1]). Let ๐‘žโ‰ฅ2 and {๐‘‹๐‘›,๐‘›โ‰ฅ1} be a sequence of ๐œŒ-mixing random variables. Assume that ๐ธ๐‘‹๐‘›=0, ๐ธ|๐‘‹๐‘›|๐‘ž<โˆž and โˆž๎“๐‘›=1๐œŒ2/๐‘ž(2๐‘›)<โˆž.(2.1) Then there exists a positive constant ๐พ=๐พ(๐‘ž,๐œŒ(โ‹…)) depending only on ๐‘ž and ๐œŒ(โ‹…) such that for any ๐‘˜โ‰ฅ0 and ๐‘›โ‰ฅ1, ๐ธ๎‚ตmax1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘˜||(๐‘–)๐‘ž๎‚ถ๎ƒฌ๎‚ตโ‰ค๐พ๐‘›max๐‘˜<๐‘–โ‰ค๐‘˜+๐‘›๐ธ๐‘‹2๐‘–๎‚ถ๐‘ž/2+๐‘›max๐‘˜<๐‘–โ‰ค๐‘˜+๐‘›๐ธ||๐‘‹๐‘–||๐‘ž๎ƒญ,(2.2) where ๐‘†๐‘˜โˆ‘(๐‘–)=๐‘˜+๐‘–๐‘—=๐‘˜+1๐‘‹๐‘—, ๐‘˜โ‰ฅ0 and ๐‘–โ‰ฅ1.

Remark 2.2. We point out that if {๐‘‹๐‘›,๐‘›โ‰ฅ1} is a sequence of ๐œŒ-mixing random variables with identical distribution and the conditions of Lemma 2.1 hold, then we have ๐ธ๎‚ตmax1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘˜||(๐‘–)๐‘ž๎‚ถ๎‚ƒ๎€ทโ‰ค๐พ๐‘›๐ธ๐‘‹21๎€ธ๐‘ž/2||๐‘‹+๐‘›๐ธ1||๐‘ž๎‚„,๐ธโŽ›โŽœโŽœโŽmax1โ‰ค๐‘–โ‰ค๐‘›|||||๐‘–๎“๐‘—=1๐‘‹๐‘—|||||๐‘žโŽžโŽŸโŽŸโŽ ๎‚ƒ๎€ทโ‰ค๐พ๐‘›๐ธ๐‘‹21๎€ธ๐‘ž/2||๐‘‹+๐‘›๐ธ1||๐‘ž๎‚„โŽกโŽขโŽขโŽฃ๎ƒฉ=๐พ๐‘›๎“๐‘—=1๐ธ๐‘‹2๐‘—๎ƒช๐‘ž/2+๐‘›๎“๐‘—=1๐ธ||๐‘‹๐‘—||๐‘žโŽคโŽฅโŽฅโŽฆ.(2.3) The inequality above is the Rosenthal type inequality of identical distributed ๐œŒ-mixing random variables.

Proof of Theorem 1.2. It is easily seen that ๐‘“(๐‘ฅ)=(๐‘Ž+๐‘ฅ)โˆ’๐›ผ is a convex function of ๐‘ฅ on [0,โˆž), therefore, we have by Jensenโ€™s inequality that ๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผโ‰ฅ๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธโˆ’๐›ผ,(2.4) which implies that liminf๐‘›โ†’โˆž๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธ๐›ผ๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผโ‰ฅ1.(2.5) To prove (1.4), it is enough to prove that limsup๐‘›โ†’โˆž๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธ๐›ผ๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผโ‰ค1.(2.6) In order to prove (2.6), we need only to show that for all ๐›ฟโˆˆ(0,1), limsup๐‘›โ†’โˆž๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธ๐›ผ๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผโ‰ค(1โˆ’๐›ฟ)โˆ’๐›ผ.(2.7) By (iii), we can see that for all ๐›ฟโˆˆ(0,1), ๐ธ๐‘1๐ผ๎€ท๐‘1>๐‘๐ต๐‘›๎€ธโ‰ค๐›ฟ2๐ธ๐‘1,๐‘›โ‰ฅ๐‘›0.(2.8) Let ๐‘ˆ๐‘›=๐ต๐‘›๐‘›โˆ’1๎“๐‘˜=1๐‘๐‘˜๐ผ๎€ท๐‘๐‘˜โ‰ค๐‘๐ต๐‘›๎€ธ,(2.9)๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผ๎€ท=๐ธ๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผ๐ผ๎€ท๐‘ˆ๐‘›โ‰ฅ๐œ‡๐‘›โˆ’๐›ฟ๐œ‡๐‘›๎€ธ๎€ท+๐ธ๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผ๐ผ๎€ท๐‘ˆ๐‘›<๐œ‡๐‘›โˆ’๐›ฟ๐œ‡๐‘›๎€ธโ‰๐‘„1+๐‘„2.(2.10) For ๐‘„1, since ๐‘‹๐‘›โ‰ฅ๐‘ˆ๐‘›, we have ๐‘„1๎€ทโ‰ค๐ธ๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผ๐ผ๎€ท๐‘‹๐‘›โ‰ฅ๐œ‡๐‘›โˆ’๐›ฟ๐œ‡๐‘›๎€ธโ‰ค๎€ท๐‘Ž+๐œ‡๐‘›โˆ’๐›ฟ๐œ‡๐‘›๎€ธโˆ’๐›ผ.(2.11) By (2.8), we have for ๐‘›โ‰ฅ๐‘›0 that ๐œ‡๐‘›โˆ’๐ธ๐‘ˆ๐‘›=๐ต๐‘›๐‘›โˆ’1๎“๐‘˜=1๐ธ๐‘๐‘˜๐ผ๎€ท๐‘๐‘˜>๐‘๐ต๐‘›๎€ธโ‰ค๐›ฟ๐œ‡๐‘›2.(2.12) Therefore, by (2.12), Markovโ€™s inequality, Remark 2.2 and ๐ถ๐‘Ÿโ€™s inequality, for any ๐‘>2 and all ๐‘› sufficiently large, ๐‘„2โ‰ค๐‘Žโˆ’๐›ผ๐‘ƒ๎€ท๐‘ˆ๐‘›<๐œ‡๐‘›โˆ’๐›ฟ๐œ‡๐‘›๎€ธ=๐‘Žโˆ’๐›ผ๐‘ƒ๎€ท๐ธ๐‘ˆ๐‘›โˆ’๐‘ˆ๐‘›>๐›ฟ๐œ‡๐‘›โˆ’๎€ท๐œ‡๐‘›โˆ’๐ธ๐‘ˆ๐‘›๎€ธ๎€ธโ‰ค๐‘Žโˆ’๐›ผ๐‘ƒ๎‚ต๐ธ๐‘ˆ๐‘›โˆ’๐‘ˆ๐‘›>๐›ฟ๐œ‡๐‘›2๎‚ถโ‰ค๐‘Žโˆ’๐›ผ๐‘ƒ๎‚ต||๐‘ˆ๐‘›โˆ’๐ธ๐‘ˆ๐‘›||>๐›ฟ๐œ‡๐‘›2๎‚ถโ‰ค๐ถ๐œ‡๐‘›โˆ’๐‘๐ธ||๐‘ˆ๐‘›โˆ’๐ธ๐‘ˆ๐‘›||๐‘โ‰ค๐ถ๐œ‡๐‘›โˆ’๐‘๎€บ๐ต๐‘›โˆ’2๐‘›๐ธ๐‘21๐ผ๎€ท๐‘1โ‰ค๐‘๐ต๐‘›๎€ธ๎€ป๐‘/2+๐ถ๐œ‡๐‘›โˆ’๐‘๎€บ๐ต๐‘›โˆ’๐‘๐‘›๐ธ๐‘๐‘1๐ผ๎€ท๐‘1โ‰ค๐‘๐ต๐‘›๎€ธ๎€ปโ‰ค๐ถ๐œ‡๐‘›โˆ’๐‘๎€บ๐ต๐‘›โˆ’1๐‘›๐ธ๐‘1๐ผ๎€ท๐‘1โ‰ค๐‘๐ต๐‘›๎€ธ๎€ป๐‘/2+๐ถ๐œ‡๐‘›โˆ’๐‘๐ต๐‘›โˆ’1๐‘›๐ธ๐‘1๐ผ๎€ท๐‘1โ‰ค๐‘๐ต๐‘›๎€ธโ‰ค๐ถ๐œ‡๐‘›โˆ’๐‘๎‚€๐œ‡๐‘›๐‘/2+๐œ‡๐‘›๎‚๎‚€๐œ‡=๐ถ๐‘›โˆ’๐‘/2+๐œ‡๐‘›โˆ’(๐‘โˆ’1)๎‚.(2.13) Taking ๐‘>max{2,2๐›ผ,๐›ผ+1}, we have by (2.10), (2.11), and (2.13) that limsup๐‘›โ†’โˆž๎€ท๐‘Ž+๐œ‡๐‘›๎€ธ๐›ผ๐ธ๎€ท๐‘Ž+๐‘‹๐‘›๎€ธโˆ’๐›ผโ‰คlimsup๐‘›โ†’โˆž๎€ท๐‘Ž+๐œ‡๐‘›๎€ธ๐›ผ๎€ท๐‘Ž+๐œ‡๐‘›โˆ’๐›ฟ๐œ‡๐‘›๎€ธโˆ’๐›ผ+limsup๐‘›โ†’โˆž๎€ท๐‘Ž+๐œ‡๐‘›๎€ธ๐›ผ๎‚ƒ๐ถ๐œ‡๐‘›โˆ’๐‘/2+๐ถ๐œ‡๐‘›โˆ’(๐‘โˆ’1)๎‚„=(1โˆ’๐›ฟ)โˆ’๐›ผ,(2.14) which implies (2.7). This completes the proof of the theorem.

Proof of Corollary 1.3. The condition ๐ต๐‘›=๐‘‚(๐‘›๐›ฟ) for some 0<๐›ฟ<1 implies that ๐œ‡๐‘›โ‰๐ธ๐‘‹๐‘›=๐ต๐‘›๐‘›โˆ’1๎“๐‘˜=1๐ธ๐‘๐‘˜=๐‘›๐ต๐‘›โˆ’1๐ธ๐‘1,(2.15) thus, ๐œ‡๐‘›โ‰ฅ๐ถ๐‘›1โˆ’๐›ฟโ†’โˆž as ๐‘›โ†’โˆž.
The fact 0<๐ธ๐‘1<โˆž and ๐ต๐‘›โ†’โˆž yield that ๐ธ๐‘1๐ผ(๐‘1>๐‘๐ต๐‘›)โ†’0 as ๐‘›โ†’โˆž, which implies that for all 0<๐œ€<1, there exists ๐‘›0>0 such that ๐ธ๐‘1๐ผ๎€ท๐‘1>๐‘๐ต๐‘›๎€ธโ‰ค๐œ€๐ธ๐‘1,๐‘›โ‰ฅ๐‘›0.(2.16) That is to say condition (iii) of Theorem 1.2 holds. Therefore, the desired result follows from Theorem 1.2 immediately.

Proof of Theorem 1.4. Firstly, we will examine Var๐‘‹๐‘›. By Remark 2.2, 0<๐ธ๐‘21<โˆž and the condition ๐ต๐‘›โ‰ฅ๐ถ๐‘›1/2 for all ๐‘› large enough, we can get that Var๐‘‹๐‘›=๐ต๐‘›โˆ’2๎ƒฉVar๐‘›๎“๐‘–=1๐‘๐‘–๎ƒชโ‰ค๐ต๐‘›โˆ’2๐ธ๎ƒฉ๐‘›๎“๐‘–=1๐‘๐‘–๎ƒช2โ‰ค๐ถ๐‘›๐ต๐‘›โˆ’2๐ธ๐‘21โ‰ค๐ถ1(2.17) for all ๐‘› large enough.
Denote ๐œ™(๐‘ฅ)=(๐‘Ž+๐‘ฅ)โˆ’๐›ผ for ๐‘ฅโ‰ฅ0. By Taylorโ€™s expansion, we can see that ๐œ™๎€ท๐‘‹๐‘›๎€ธ๎€ท=๐œ™๐ธ๐‘‹๐‘›๎€ธ+๐œ™๎…ž๎€ท๐œ‰๐‘›๐‘‹๎€ธ๎€ท๐‘›โˆ’๐ธ๐‘‹๐‘›๎€ธ,(2.18) where ๐œ‰๐‘› is between ๐‘‹๐‘› and ๐ธ๐‘‹๐‘›. It is easily seen that {๐œ™๎…ž(๐‘ฅ)}2 is decreasing in ๐‘ฅโ‰ฅ0. Therefore, by (2.18), Cauchy-Schwartz inequality, (2.17) and (1.4), we have ๎€บ๎€ท๐‘‹๐ธ๐œ™๐‘›๎€ธ๎€ทโˆ’๐œ™๐ธ๐‘‹๐‘›๎€ธ๎€ป2๎€บ๐œ™=๐ธ๎…ž๎€ท๐œ‰๐‘›๐‘‹๎€ธ๎€ท๐‘›โˆ’๐ธ๐‘‹๐‘›๎€ธ๎€ป2๎€บ๐œ™โ‰ค๐ธ๎…ž๎€ท๐œ‰๐‘›๎€ธ๎€ป2Var๐‘‹๐‘›โ‰ค๐ถ1๐ธ๎€บ๐œ™๎…ž๎€ท๐œ‰๐‘›๎€ธ๎€ป2=๐ถ1๐ธ๎€บ๐œ™๎…ž๎€ท๐œ‰๐‘›๎€ธ๎€ป2๐ผ๎€ท๐‘‹๐‘›โ‰ค๐ธ๐‘‹๐‘›๎€ธ+๐ถ1๐ธ๎€บ๐œ™๎…ž๎€ท๐œ‰๐‘›๎€ธ๎€ป2๐ผ๎€ท๐‘‹๐‘›>๐ธ๐‘‹๐‘›๎€ธโ‰ค๐ถ1๐ธ๎€บ๐œ™๎…ž๎€ท๐‘‹๐‘›๎€ธ๎€ป2+๐ถ1๐ธ๎€บ๐œ™๎…ž๎€ท๐ธ๐‘‹๐‘›๎€ธ๎€ป2โˆผ2๐ถ1๐ธ๎€บ๐œ™๎…ž๎€ท๐ธ๐‘‹๐‘›๎€ธ๎€ป2=2๐ถ1๐›ผ2๎€ท๐‘Ž+๐ธ๐‘‹๐‘›๎€ธโˆ’2(๐›ผ+1).(2.19) This leads to (1.8). The proof is complete.

Proof of Theorem 1.5. The proof is similar to that of Theorem 1.4. In place of Var๐‘‹๐‘›โ‰ค๐ถ1, we make the use of Var๐‘‹๐‘›โ‰ค๐ถ๐‘›๐ต๐‘›โˆ’2๐ธ๐‘21โ‰๐ถ2๐‘›๐ต๐‘›โˆ’2. The proof is complete.

Acknowledgments

The authors are most grateful to the Editor Tetsuji Tokihiro and an anonymous referee for the careful reading of the paper and valuable suggestions which helped to improve an earlier version of this paper. The paper is supported by the Academic innovation team of Anhui University (KJTD001B).

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