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 šœŒ-mixing random variables. In the paper, we will further study the asymptotic approximation of inverse moments for nonnegative š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 šœŒ-mixing random variables.

Let {š‘‹š‘›,š‘›ā‰„1} be a sequence of random variables defined on a fixed probability space (Ī©,ā„±,š‘ƒ). Let š‘› and š‘š be positive integers. Write ā„±š‘šš‘›=šœŽ(š‘‹š‘–,š‘›ā‰¤š‘–ā‰¤š‘š) and ā„±š’®=šœŽ(š‘‹š‘–,š‘–āˆˆš‘†āŠ‚ā„•). Given šœŽ-algebras ā„¬,ā„› in ā„±, let šœŒ(ā„¬,ā„›)=supš‘‹āˆˆšæ2(ā„¬),š‘Œāˆˆšæ2(ā„›)||||šøš‘‹š‘Œāˆ’šøš‘‹šøš‘Œāˆš.Var(š‘‹)ā‹…Var(š‘Œ)(1.1) Define the šœŒ-mixing coefficients by šœŒ(š‘›)=supš‘˜ā‰„1šœŒī€·ā„±š‘˜1,ā„±āˆžš‘˜+š‘›ī€ø,š‘›ā‰„0.(1.2)

Definition 1.1. A sequence {š‘‹š‘›,š‘›ā‰„1} of random variables is said to be šœŒ-mixing if šœŒ(š‘›)ā†“0 as š‘›ā†’āˆž.

šœŒ-mixing sequence was introduced by Kolmogorov and Rozanov [1]. It is easily seen that š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 šœŒ-mixing random variables with identical distribution.

Let {š‘š‘›,š‘›ā‰„1} be a sequence of independent nonnegative random variables with finite second moments. Denote š‘‹š‘›=āˆ‘š‘›š‘–=1š‘š‘–šµš‘›,šµ2š‘›=š‘›ī“š‘–=1Varš‘š‘–.(1.3) It is interesting to show that under suitable conditions the following equivalence relation holds, namely, šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š‘Ÿāˆ¼ī€·š‘Ž+šøš‘‹š‘›ī€øāˆ’š‘Ÿ,š‘›āŸ¶āˆž,(1.4) where š‘Ž>0 and š‘Ÿ>0 are arbitrary real numbers.

Here and below, for two positive sequences {š‘š‘›,š‘›ā‰„1} and {š‘‘š‘›,š‘›ā‰„1}, we write š‘š‘›āˆ¼š‘‘š‘› if š‘š‘›š‘‘š‘›āˆ’1ā†’1 as š‘›ā†’āˆž. š¶ 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 Theoremā€‰ā€‰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)š‘Ÿ<3 (š‘Ÿ<4, in the i.i.d. case);(ii)šøš‘‹š‘›ā†’āˆž, šøš‘3š‘›<āˆž;(iii)(šæš‘ condition) āˆ‘š‘›š‘–=1šø|š‘š‘–āˆ’šøš‘š‘–|š‘/šµš‘š‘›ā†’0 (š‘=3).

Hu et al. [12] considered weaker conditions: šøš‘š‘›2+š›æ<āˆž, where š‘š‘› satisfies šæ2+š›æ condition and 0<š›æā‰¤1. Wu et al. [13] applied Bernsteinā€™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 šµš‘›=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 šœŒ-mixing random variables and obtained the following result.

Theorem A. Let {š‘š‘›,š‘›ā‰„1} be a nonnegative šœŒ-mixing sequence with āˆ‘āˆžš‘›=1šœŒ(š‘›)<āˆž. Suppose that (i)šøš‘2š‘›<āˆž, for all š‘›ā‰„1;(ii)šøš‘‹š‘›ā†’āˆž, where š‘‹š‘› is defined by (1.3);(iii)for some šœ‚>0,š‘…š‘›(šœ‚)āˆ¶=šµš‘›š‘›āˆ’2ī“š‘–=1šøš‘2š‘–š¼ī€·š‘š‘–>šœ‚šµš‘›ī€øāŸ¶0,š‘›āŸ¶āˆž;(1.5)(iv)for some š‘”āˆˆ(0,1) and any positive constants š‘Ž,š‘Ÿ,š¶,limš‘›ā†’āˆžī€·š‘Ž+šøš‘‹š‘›ī€øš‘ŸīƒÆī€·ā‹…expāˆ’š¶ā‹…šøš‘‹š‘›ī€øš‘”š‘›īƒ°=0.(1.6) Then for any š‘Ž>0 and š‘Ÿ>0, (1.4) holds.

In this paper, we will further study the asymptotic approximation of inverse moments for nonnegative š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 šœŒ-mixing random variables. Our main results are as follows.

Theorem 1.2. Let {š‘š‘›,š‘›ā‰„1} be a sequence of nonnegative šœŒ-mixing random variables with identical distribution and let {šµš‘›,š‘›ā‰„1} be a sequence of positive constants. Let š‘Ž>0 and š›¼>0 be real numbers. š‘>max{2,2š›¼,š›¼+1}. Assume that āˆ‘āˆžš‘›=1šœŒ2/š‘(2š‘›)<āˆž. Suppose that (i)0<šøš‘š‘›<āˆž, for all š‘›ā‰„1;(ii)šœ‡š‘›ā‰šøš‘‹š‘›ā†’āˆž as š‘›ā†’āˆž, where š‘‹š‘›=šµš‘›āˆ’1āˆ‘š‘›š‘˜=1š‘š‘˜;(iii)for all 0<šœ€<1, there exist š‘>0 and š‘›0>0 such that šøš‘1š¼ī€·š‘1>š‘šµš‘›ī€øā‰¤šœ€šøš‘1,š‘›ā‰„š‘›0.(1.7) Then (1.4) holds.

Corollary 1.3. Let {š‘š‘›,š‘›ā‰„1} be a sequence of nonnegative šœŒ-mixing random variables with identical distribution and 0<šøš‘1<āˆž. Let {šµš‘›,š‘›ā‰„1} be a sequence of positive constants satisfying šµš‘›=š‘‚(š‘›š›æ) for some 0<š›æ<1 and šµš‘›ā†’āˆž as š‘›ā†’āˆž. Let š‘Ž>0 and š›¼>0 be real numbers. š‘>max{2,2š›¼,š›¼+1}. Assume that āˆ‘āˆžš‘›=1šœŒ2/š‘(2š‘›)<āˆž. 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<šøš‘2š‘›<āˆž. š‘>max{2,4(š›¼+1),2š›¼+3}. If šµš‘›ā‰„š¶š‘›1/2 for all š‘› large enough, where š¶ is a positive constant, then ||ī€·š‘Ž+šøš‘‹š‘›ī€øš›¼šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š›¼||ī‚€ī€·āˆ’1=š‘‚š‘Ž+šøš‘‹š‘›ī€øāˆ’1ī‚.(1.8)

Theorem 1.5. Assume that conditions of Theorem 1.2 are satisfied and 0<šøš‘2š‘›<āˆž. š‘>max{2,4(š›¼+1),2š›¼+3}. Then ||ī€·š‘Ž+šøš‘‹š‘›ī€øš›¼šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š›¼||ī€·š‘›āˆ’1=š‘‚āˆ’1/2ī€ø.(1.9)

Taking šµš‘›ā‰”1 in Theorem 1.2, we have the following asymptotic approximation of inverse moments for the partial sums of nonnegative šœŒ-mixing random variables with identical distribution.

Theorem 1.6. Let {š‘š‘›,š‘›ā‰„1} be a sequence of nonnegative šœŒ-mixing random variables with identical distribution. Let š‘Ž>0 and š›¼>0 be real numbers. š‘>max{2,2š›¼,š›¼+1}. Assume that āˆ‘āˆžš‘›=1šœŒ2/š‘(2š‘›)<āˆž. Suppose that (i)0<šøš‘š‘›<āˆž, āˆ€š‘›ā‰„1;(ii)šœˆš‘›ā‰šøš‘Œš‘›ā†’āˆž as š‘›ā†’āˆž, where š‘Œš‘›=āˆ‘š‘›š‘˜=1š‘š‘˜;(iii)for all 0<šœ€<1, there exist š‘>0 and š‘›0>0 such that šøš‘1š¼ī€·š‘1ī€ø>š‘ā‰¤šœ€šøš‘1,š‘›ā‰„š‘›0.(1.10) Then šø(š‘Ž+š‘Œš‘›)āˆ’š›¼āˆ¼(š‘Ž+šøš‘Œš‘›)āˆ’š›¼.

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 šøš‘š‘›<āˆž, for all š‘›ā‰„1, which is weaker than the condition šøš‘2š‘›<āˆž, for all š‘›ā‰„1 in the above cited references. Secondly, {šµš‘›,š‘›ā‰„1} is an arbitrary sequence of positive constants in Theorem 1.2, while šµ2š‘›=āˆ‘š‘›š‘–=1Varš‘š‘– 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 šµš‘›š‘›āˆ’1ī“š‘–=1šøš‘š‘–š¼ī€·š‘š‘–>šœ‚šµš‘›ī€øā‰¤šœ‚āˆ’1šµš‘›š‘›āˆ’2ī“š‘–=1šøš‘2š‘–š¼ī€·š‘š‘–>šœ‚šµš‘›ī€øāŸ¶0,š‘›āŸ¶āˆž,(1.11) which implies that for all 0<šœ€<1, there exists a positive integer š‘›0 such that šµš‘›š‘›āˆ’1ī“š‘–=1šøš‘š‘–š¼ī€·š‘š‘–>šœ‚šµš‘›ī€øā‰¤šœ€šœ‡š‘›=šœ€šµš‘›š‘›āˆ’1ī“š‘–=1šøš‘š‘–,š‘›ā‰„š‘›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 šœŒ-mixing random variables.

Lemma 2.1 (c.f. Shao [18, Corollaryā€‰ā€‰1.1]). Let š‘žā‰„2 and {š‘‹š‘›,š‘›ā‰„1} be a sequence of šœŒ-mixing random variables. Assume that šøš‘‹š‘›=0, šø|š‘‹š‘›|š‘ž<āˆž and āˆžī“š‘›=1šœŒ2/š‘ž(2š‘›)<āˆž.(2.1) Then there exists a positive constant š¾=š¾(š‘ž,šœŒ(ā‹…)) depending only on š‘ž and šœŒ(ā‹…) such that for any š‘˜ā‰„0 and š‘›ā‰„1, šøī‚µmax1ā‰¤š‘–ā‰¤š‘›||š‘†š‘˜||(š‘–)š‘žī‚¶īƒ¬ī‚µā‰¤š¾š‘›maxš‘˜<š‘–ā‰¤š‘˜+š‘›šøš‘‹2š‘–ī‚¶š‘ž/2+š‘›maxš‘˜<š‘–ā‰¤š‘˜+š‘›šø||š‘‹š‘–||š‘žīƒ­,(2.2) where š‘†š‘˜āˆ‘(š‘–)=š‘˜+š‘–š‘—=š‘˜+1š‘‹š‘—, š‘˜ā‰„0 and š‘–ā‰„1.

Remark 2.2. We point out that if {š‘‹š‘›,š‘›ā‰„1} is a sequence of šœŒ-mixing random variables with identical distribution and the conditions of Lemma 2.1 hold, then we have šøī‚µmax1ā‰¤š‘–ā‰¤š‘›||š‘†š‘˜||(š‘–)š‘žī‚¶ī‚ƒī€·ā‰¤š¾š‘›šøš‘‹21ī€øš‘ž/2||š‘‹+š‘›šø1||š‘žī‚„,šøāŽ›āŽœāŽœāŽmax1ā‰¤š‘–ā‰¤š‘›|||||š‘–ī“š‘—=1š‘‹š‘—|||||š‘žāŽžāŽŸāŽŸāŽ ī‚ƒī€·ā‰¤š¾š‘›šøš‘‹21ī€øš‘ž/2||š‘‹+š‘›šø1||š‘žī‚„āŽ”āŽ¢āŽ¢āŽ£īƒ©=š¾š‘›ī“š‘—=1šøš‘‹2š‘—īƒŖš‘ž/2+š‘›ī“š‘—=1šø||š‘‹š‘—||š‘žāŽ¤āŽ„āŽ„āŽ¦.(2.3) The inequality above is the Rosenthal type inequality of identical distributed šœŒ-mixing random variables.

Proof of Theorem 1.2. It is easily seen that š‘“(š‘„)=(š‘Ž+š‘„)āˆ’š›¼ is a convex function of š‘„ on [0,āˆž), therefore, we have by Jensenā€™s inequality that šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š›¼ā‰„ī€·š‘Ž+šøš‘‹š‘›ī€øāˆ’š›¼,(2.4) which implies that liminfš‘›ā†’āˆžī€·š‘Ž+šøš‘‹š‘›ī€øš›¼šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š›¼ā‰„1.(2.5) To prove (1.4), it is enough to prove that limsupš‘›ā†’āˆžī€·š‘Ž+šøš‘‹š‘›ī€øš›¼šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š›¼ā‰¤1.(2.6) In order to prove (2.6), we need only to show that for all š›æāˆˆ(0,1), limsupš‘›ā†’āˆžī€·š‘Ž+šøš‘‹š‘›ī€øš›¼šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š›¼ā‰¤(1āˆ’š›æ)āˆ’š›¼.(2.7) By (iii), we can see that for all š›æāˆˆ(0,1), šøš‘1š¼ī€·š‘1>š‘šµš‘›ī€øā‰¤š›æ2šøš‘1,š‘›ā‰„š‘›0.(2.8) Let š‘ˆš‘›=šµš‘›š‘›āˆ’1ī“š‘˜=1š‘š‘˜š¼ī€·š‘š‘˜ā‰¤š‘šµš‘›ī€ø,(2.9)šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š›¼ī€·=šøš‘Ž+š‘‹š‘›ī€øāˆ’š›¼š¼ī€·š‘ˆš‘›ā‰„šœ‡š‘›āˆ’š›æšœ‡š‘›ī€øī€·+šøš‘Ž+š‘‹š‘›ī€øāˆ’š›¼š¼ī€·š‘ˆš‘›<šœ‡š‘›āˆ’š›æšœ‡š‘›ī€øā‰š‘„1+š‘„2.(2.10) For š‘„1, since š‘‹š‘›ā‰„š‘ˆš‘›, we have š‘„1ī€·ā‰¤šøš‘Ž+š‘‹š‘›ī€øāˆ’š›¼š¼ī€·š‘‹š‘›ā‰„šœ‡š‘›āˆ’š›æšœ‡š‘›ī€øā‰¤ī€·š‘Ž+šœ‡š‘›āˆ’š›æšœ‡š‘›ī€øāˆ’š›¼.(2.11) By (2.8), we have for š‘›ā‰„š‘›0 that šœ‡š‘›āˆ’šøš‘ˆš‘›=šµš‘›š‘›āˆ’1ī“š‘˜=1šøš‘š‘˜š¼ī€·š‘š‘˜>š‘šµš‘›ī€øā‰¤š›æšœ‡š‘›2.(2.12) Therefore, by (2.12), Markovā€™s inequality, Remark 2.2 and š¶š‘Ÿā€™s inequality, for any š‘>2 and all š‘› sufficiently large, š‘„2ā‰¤š‘Žāˆ’š›¼š‘ƒī€·š‘ˆš‘›<šœ‡š‘›āˆ’š›æšœ‡š‘›ī€ø=š‘Žāˆ’š›¼š‘ƒī€·šøš‘ˆš‘›āˆ’š‘ˆš‘›>š›æšœ‡š‘›āˆ’ī€·šœ‡š‘›āˆ’šøš‘ˆš‘›ī€øī€øā‰¤š‘Žāˆ’š›¼š‘ƒī‚µšøš‘ˆš‘›āˆ’š‘ˆš‘›>š›æšœ‡š‘›2ī‚¶ā‰¤š‘Žāˆ’š›¼š‘ƒī‚µ||š‘ˆš‘›āˆ’šøš‘ˆš‘›||>š›æšœ‡š‘›2ī‚¶ā‰¤š¶šœ‡š‘›āˆ’š‘šø||š‘ˆš‘›āˆ’šøš‘ˆš‘›||š‘ā‰¤š¶šœ‡š‘›āˆ’š‘ī€ŗšµš‘›āˆ’2š‘›šøš‘21š¼ī€·š‘1ā‰¤š‘šµš‘›ī€øī€»š‘/2+š¶šœ‡š‘›āˆ’š‘ī€ŗšµš‘›āˆ’š‘š‘›šøš‘š‘1š¼ī€·š‘1ā‰¤š‘šµš‘›ī€øī€»ā‰¤š¶šœ‡š‘›āˆ’š‘ī€ŗšµš‘›āˆ’1š‘›šøš‘1š¼ī€·š‘1ā‰¤š‘šµš‘›ī€øī€»š‘/2+š¶šœ‡š‘›āˆ’š‘šµš‘›āˆ’1š‘›šøš‘1š¼ī€·š‘1ā‰¤š‘šµš‘›ī€øā‰¤š¶šœ‡š‘›āˆ’š‘ī‚€šœ‡š‘›š‘/2+šœ‡š‘›ī‚ī‚€šœ‡=š¶š‘›āˆ’š‘/2+šœ‡š‘›āˆ’(š‘āˆ’1)ī‚.(2.13) Taking š‘>max{2,2š›¼,š›¼+1}, we have by (2.10), (2.11), and (2.13) that limsupš‘›ā†’āˆžī€·š‘Ž+šœ‡š‘›ī€øš›¼šøī€·š‘Ž+š‘‹š‘›ī€øāˆ’š›¼ā‰¤limsupš‘›ā†’āˆžī€·š‘Ž+šœ‡š‘›ī€øš›¼ī€·š‘Ž+šœ‡š‘›āˆ’š›æšœ‡š‘›ī€øāˆ’š›¼+limsupš‘›ā†’āˆžī€·š‘Ž+šœ‡š‘›ī€øš›¼ī‚ƒš¶šœ‡š‘›āˆ’š‘/2+š¶šœ‡š‘›āˆ’(š‘āˆ’1)ī‚„=(1āˆ’š›æ)āˆ’š›¼,(2.14) which implies (2.7). This completes the proof of the theorem.

Proof of Corollary 1.3. The condition šµš‘›=š‘‚(š‘›š›æ) for some 0<š›æ<1 implies that šœ‡š‘›ā‰šøš‘‹š‘›=šµš‘›š‘›āˆ’1ī“š‘˜=1šøš‘š‘˜=š‘›šµš‘›āˆ’1šøš‘1,(2.15) thus, šœ‡š‘›ā‰„š¶š‘›1āˆ’š›æā†’āˆž as š‘›ā†’āˆž.
The fact 0<šøš‘1<āˆž and šµš‘›ā†’āˆž yield that šøš‘1š¼(š‘1>š‘šµš‘›)ā†’0 as š‘›ā†’āˆž, which implies that for all 0<šœ€<1, there exists š‘›0>0 such that šøš‘1š¼ī€·š‘1>š‘šµš‘›ī€øā‰¤šœ€šøš‘1,š‘›ā‰„š‘›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 Varš‘‹š‘›. By Remark 2.2, 0<šøš‘21<āˆž and the condition šµš‘›ā‰„š¶š‘›1/2 for all š‘› large enough, we can get that Varš‘‹š‘›=šµš‘›āˆ’2īƒ©Varš‘›ī“š‘–=1š‘š‘–īƒŖā‰¤šµš‘›āˆ’2šøīƒ©š‘›ī“š‘–=1š‘š‘–īƒŖ2ā‰¤š¶š‘›šµš‘›āˆ’2šøš‘21ā‰¤š¶1(2.17) for all š‘› large enough.
Denote šœ™(š‘„)=(š‘Ž+š‘„)āˆ’š›¼ for š‘„ā‰„0. By Taylorā€™s expansion, we can see that šœ™ī€·š‘‹š‘›ī€øī€·=šœ™šøš‘‹š‘›ī€ø+šœ™ī…žī€·šœ‰š‘›š‘‹ī€øī€·š‘›āˆ’šøš‘‹š‘›ī€ø,(2.18) where šœ‰š‘› is between š‘‹š‘› and šøš‘‹š‘›. It is easily seen that {šœ™ī…ž(š‘„)}2 is decreasing in š‘„ā‰„0. Therefore, by (2.18), Cauchy-Schwartz inequality, (2.17) and (1.4), we have ī€ŗī€·š‘‹šøšœ™š‘›ī€øī€·āˆ’šœ™šøš‘‹š‘›ī€øī€»2ī€ŗšœ™=šøī…žī€·šœ‰š‘›š‘‹ī€øī€·š‘›āˆ’šøš‘‹š‘›ī€øī€»2ī€ŗšœ™ā‰¤šøī…žī€·šœ‰š‘›ī€øī€»2Varš‘‹š‘›ā‰¤š¶1šøī€ŗšœ™ī…žī€·šœ‰š‘›ī€øī€»2=š¶1šøī€ŗšœ™ī…žī€·šœ‰š‘›ī€øī€»2š¼ī€·š‘‹š‘›ā‰¤šøš‘‹š‘›ī€ø+š¶1šøī€ŗšœ™ī…žī€·šœ‰š‘›ī€øī€»2š¼ī€·š‘‹š‘›>šøš‘‹š‘›ī€øā‰¤š¶1šøī€ŗšœ™ī…žī€·š‘‹š‘›ī€øī€»2+š¶1šøī€ŗšœ™ī…žī€·šøš‘‹š‘›ī€øī€»2āˆ¼2š¶1šøī€ŗšœ™ī…žī€·šøš‘‹š‘›ī€øī€»2=2š¶1š›¼2ī€·š‘Ž+šøš‘‹š‘›ī€øāˆ’2(š›¼+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 Varš‘‹š‘›ā‰¤š¶1, we make the use of Varš‘‹š‘›ā‰¤š¶š‘›šµš‘›āˆ’2šøš‘21ā‰š¶2š‘›šµš‘›āˆ’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).