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

Let {𝑋𝑛,𝑛≥1} be a sequence of random variables defined on a fixed probability space (Ω,ℱ,𝑃). Let 𝑛 and 𝑚 be positive integers. Write ℱ𝑚𝑛=ğœŽ(𝑋𝑖,𝑛≤𝑖≤𝑚) and ℱ𝒮=ğœŽ(𝑋𝑖,𝑖∈𝑆⊂ℕ). Given ğœŽ-algebras ℬ,ℛ in ℱ, let 𝜌(ℬ,ℛ)=sup𝑋∈𝐿2(ℬ),𝑌∈𝐿2(ℛ)||||𝐸𝑋𝑌−𝐸𝑋𝐸𝑌√.Var(𝑋)⋅Var(𝑌)(1.1) Define the 𝜌-mixing coefficients by 𝜌(𝑛)=sup𝑘≥1𝜌ℱ𝑘1,â„±âˆžğ‘˜+𝑛,𝑛≥0.(1.2)

Definition 1.1. A sequence {𝑋𝑛,𝑛≥1} of random variables is said to be 𝜌-mixing if 𝜌(𝑛)↓0 as ğ‘›â†’âˆž.

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

Let {𝑍𝑛,𝑛≥1} be a sequence of independent nonnegative random variables with finite second moments. Denote 𝑋𝑛=∑𝑛𝑖=1𝑍𝑖𝐵𝑛,𝐵2𝑛=𝑛𝑖=1Var𝑍𝑖.(1.3) It is interesting to show that under suitable conditions the following equivalence relation holds, namely, ğ¸î€·ğ‘Ž+ğ‘‹ğ‘›î€¸âˆ’ğ‘Ÿâˆ¼î€·ğ‘Ž+𝐸𝑋𝑛−𝑟,ğ‘›âŸ¶âˆž,(1.4) where ğ‘Ž>0 and 𝑟>0 are arbitrary real numbers.

Here and below, for two positive sequences {𝑐𝑛,𝑛≥1} and {𝑑𝑛,𝑛≥1}, we write 𝑐𝑛∼𝑑𝑛 if 𝑐𝑛𝑑𝑛−1→1 as ğ‘›â†’âˆž. 𝐶 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 Theorem  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)𝑟<3 (𝑟<4, in the i.i.d. case);(ii)ğ¸ğ‘‹ğ‘›â†’âˆž, 𝐸𝑍3𝑛<∞;(iii)(𝐿𝑐 condition) ∑𝑛𝑖=1𝐸|𝑍𝑖−𝐸𝑍𝑖|𝑐/𝐵𝑐𝑛→0 (𝑐=3).

Hu et al. [12] considered weaker conditions: 𝐸𝑍𝑛2+𝛿<∞, where 𝑍𝑛 satisfies 𝐿2+𝛿 condition and 0<𝛿≤1. Wu et al. [13] applied Bernstein’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 𝐵𝑛=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 𝜌-mixing random variables and obtained the following result.

Theorem A. Let {𝑍𝑛,𝑛≥1} be a nonnegative 𝜌-mixing sequence with âˆ‘âˆžğ‘›=1𝜌(𝑛)<∞. Suppose that (i)𝐸𝑍2𝑛<∞, for all 𝑛≥1;(ii)ğ¸ğ‘‹ğ‘›â†’âˆž, where 𝑋𝑛 is defined by (1.3);(iii)for some 𝜂>0,𝑅𝑛(𝜂)∶=𝐵𝑛𝑛−2𝑖=1𝐸𝑍2𝑖𝐼𝑍𝑖>𝜂𝐵𝑛⟶0,ğ‘›âŸ¶âˆž;(1.5)(iv)for some 𝑡∈(0,1) and any positive constants ğ‘Ž,𝑟,𝐶,limğ‘›â†’âˆžî€·ğ‘Ž+𝐸𝑋𝑛𝑟⋅exp−𝐶⋅𝐸𝑋𝑛𝑡𝑛=0.(1.6) Then for any ğ‘Ž>0 and 𝑟>0, (1.4) holds.

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

Theorem 1.2. Let {𝑍𝑛,𝑛≥1} be a sequence of nonnegative 𝜌-mixing random variables with identical distribution and let {𝐵𝑛,𝑛≥1} be a sequence of positive constants. Let ğ‘Ž>0 and 𝛼>0 be real numbers. 𝑝>max{2,2𝛼,𝛼+1}. Assume that âˆ‘âˆžğ‘›=1𝜌2/𝑝(2𝑛)<∞. Suppose that (i)0<𝐸𝑍𝑛<∞, for all 𝑛≥1;(ii)ğœ‡ğ‘›â‰ğ¸ğ‘‹ğ‘›â†’âˆž as ğ‘›â†’âˆž, where 𝑋𝑛=𝐵𝑛−1∑𝑛𝑘=1𝑍𝑘;(iii)for all 0<𝜀<1, there exist 𝑏>0 and 𝑛0>0 such that 𝐸𝑍1𝐼𝑍1>𝑏𝐵𝑛≤𝜀𝐸𝑍1,𝑛≥𝑛0.(1.7) Then (1.4) holds.

Corollary 1.3. Let {𝑍𝑛,𝑛≥1} be a sequence of nonnegative 𝜌-mixing random variables with identical distribution and 0<𝐸𝑍1<∞. Let {𝐵𝑛,𝑛≥1} be a sequence of positive constants satisfying 𝐵𝑛=𝑂(𝑛𝛿) for some 0<𝛿<1 and ğµğ‘›â†’âˆž as ğ‘›â†’âˆž. Let ğ‘Ž>0 and 𝛼>0 be real numbers. 𝑝>max{2,2𝛼,𝛼+1}. Assume that âˆ‘âˆžğ‘›=1𝜌2/𝑝(2𝑛)<∞. 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<𝐸𝑍2𝑛<∞. 𝑝>max{2,4(𝛼+1),2𝛼+3}. If 𝐵𝑛≥𝐶𝑛1/2 for all 𝑛 large enough, where 𝐶 is a positive constant, then ||î€·ğ‘Ž+ğ¸ğ‘‹ğ‘›î€¸ğ›¼ğ¸î€·ğ‘Ž+𝑋𝑛−𝛼||−1=ğ‘‚ğ‘Ž+𝐸𝑋𝑛−1.(1.8)

Theorem 1.5. Assume that conditions of Theorem 1.2 are satisfied and 0<𝐸𝑍2𝑛<∞. 𝑝>max{2,4(𝛼+1),2𝛼+3}. Then ||î€·ğ‘Ž+ğ¸ğ‘‹ğ‘›î€¸ğ›¼ğ¸î€·ğ‘Ž+𝑋𝑛−𝛼||𝑛−1=𝑂−1/2.(1.9)

Taking 𝐵𝑛≡1 in Theorem 1.2, we have the following asymptotic approximation of inverse moments for the partial sums of nonnegative 𝜌-mixing random variables with identical distribution.

Theorem 1.6. Let {𝑍𝑛,𝑛≥1} be a sequence of nonnegative 𝜌-mixing random variables with identical distribution. Let ğ‘Ž>0 and 𝛼>0 be real numbers. 𝑝>max{2,2𝛼,𝛼+1}. Assume that âˆ‘âˆžğ‘›=1𝜌2/𝑝(2𝑛)<∞. Suppose that (i)0<𝐸𝑍𝑛<∞, ∀𝑛≥1;(ii)ğœˆğ‘›â‰ğ¸ğ‘Œğ‘›â†’âˆž as ğ‘›â†’âˆž, where 𝑌𝑛=∑𝑛𝑘=1𝑍𝑘;(iii)for all 0<𝜀<1, there exist 𝑏>0 and 𝑛0>0 such that 𝐸𝑍1𝐼𝑍1>𝑏≤𝜀𝐸𝑍1,𝑛≥𝑛0.(1.10) Then 𝐸(ğ‘Ž+𝑌𝑛)−𝛼∼(ğ‘Ž+𝐸𝑌𝑛)−𝛼.

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 𝐸𝑍𝑛<∞, for all 𝑛≥1, which is weaker than the condition 𝐸𝑍2𝑛<∞, for all 𝑛≥1 in the above cited references. Secondly, {𝐵𝑛,𝑛≥1} is an arbitrary sequence of positive constants in Theorem 1.2, while 𝐵2𝑛=∑𝑛𝑖=1Var𝑍𝑖 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 𝐵𝑛𝑛−1𝑖=1𝐸𝑍𝑖𝐼𝑍𝑖>𝜂𝐵𝑛≤𝜂−1𝐵𝑛𝑛−2𝑖=1𝐸𝑍2𝑖𝐼𝑍𝑖>𝜂𝐵𝑛⟶0,ğ‘›âŸ¶âˆž,(1.11) which implies that for all 0<𝜀<1, there exists a positive integer 𝑛0 such that 𝐵𝑛𝑛−1𝑖=1𝐸𝑍𝑖𝐼𝑍𝑖>𝜂𝐵𝑛≤𝜀𝜇𝑛=𝜀𝐵𝑛𝑛−1𝑖=1𝐸𝑍𝑖,𝑛≥𝑛0,(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 𝜌-mixing random variables.

Lemma 2.1 (c.f. Shao [18, Corollary  1.1]). Let ğ‘žâ‰¥2 and {𝑋𝑛,𝑛≥1} be a sequence of 𝜌-mixing random variables. Assume that 𝐸𝑋𝑛=0, 𝐸|𝑋𝑛|ğ‘ž<∞ and âˆžî“ğ‘›=1𝜌2/ğ‘ž(2𝑛)<∞.(2.1) Then there exists a positive constant 𝐾=𝐾(ğ‘ž,𝜌(⋅)) depending only on ğ‘ž and 𝜌(⋅) such that for any 𝑘≥0 and 𝑛≥1, 𝐸max1≤𝑖≤𝑛||𝑆𝑘||(𝑖)ğ‘žî‚¶îƒ¬î‚µâ‰¤ğ¾ğ‘›max𝑘<𝑖≤𝑘+𝑛𝐸𝑋2ğ‘–î‚¶ğ‘ž/2+𝑛max𝑘<𝑖≤𝑘+𝑛𝐸||𝑋𝑖||ğ‘žîƒ­,(2.2) where 𝑆𝑘∑(𝑖)=𝑘+𝑖𝑗=𝑘+1𝑋𝑗, 𝑘≥0 and 𝑖≥1.

Remark 2.2. We point out that if {𝑋𝑛,𝑛≥1} is a sequence of 𝜌-mixing random variables with identical distribution and the conditions of Lemma 2.1 hold, then we have 𝐸max1≤𝑖≤𝑛||𝑆𝑘||(𝑖)ğ‘žî‚¶î‚ƒî€·â‰¤ğ¾ğ‘›ğ¸ğ‘‹21î€¸ğ‘ž/2||𝑋+𝑛𝐸1||ğ‘žî‚„,ğ¸âŽ›âŽœâŽœâŽmax1≤𝑖≤𝑛|||||𝑖𝑗=1𝑋𝑗|||||ğ‘žâŽžâŽŸâŽŸâŽ î‚ƒî€·â‰¤ğ¾ğ‘›ğ¸ğ‘‹21î€¸ğ‘ž/2||𝑋+𝑛𝐸1||ğ‘žî‚„âŽ¡âŽ¢âŽ¢âŽ£îƒ©=𝐾𝑛𝑗=1𝐸𝑋2ğ‘—îƒªğ‘ž/2+𝑛𝑗=1𝐸||𝑋𝑗||ğ‘žâŽ¤âŽ¥âŽ¥âŽ¦.(2.3) The inequality above is the Rosenthal type inequality of identical distributed 𝜌-mixing random variables.

Proof of Theorem 1.2. It is easily seen that 𝑓(𝑥)=(ğ‘Ž+𝑥)−𝛼 is a convex function of 𝑥 on [0,∞), therefore, we have by Jensen’s inequality that ğ¸î€·ğ‘Ž+ğ‘‹ğ‘›î€¸âˆ’ğ›¼â‰¥î€·ğ‘Ž+𝐸𝑋𝑛−𝛼,(2.4) which implies that liminfğ‘›â†’âˆžî€·ğ‘Ž+ğ¸ğ‘‹ğ‘›î€¸ğ›¼ğ¸î€·ğ‘Ž+𝑋𝑛−𝛼≥1.(2.5) To prove (1.4), it is enough to prove that limsupğ‘›â†’âˆžî€·ğ‘Ž+ğ¸ğ‘‹ğ‘›î€¸ğ›¼ğ¸î€·ğ‘Ž+𝑋𝑛−𝛼≤1.(2.6) In order to prove (2.6), we need only to show that for all 𝛿∈(0,1), limsupğ‘›â†’âˆžî€·ğ‘Ž+ğ¸ğ‘‹ğ‘›î€¸ğ›¼ğ¸î€·ğ‘Ž+𝑋𝑛−𝛼≤(1−𝛿)−𝛼.(2.7) By (iii), we can see that for all 𝛿∈(0,1), 𝐸𝑍1𝐼𝑍1>𝑏𝐵𝑛≤𝛿2𝐸𝑍1,𝑛≥𝑛0.(2.8) Let 𝑈𝑛=𝐵𝑛𝑛−1𝑘=1𝑍𝑘𝐼𝑍𝑘≤𝑏𝐵𝑛,(2.9)ğ¸î€·ğ‘Ž+𝑋𝑛−𝛼=ğ¸ğ‘Ž+𝑋𝑛−𝛼𝐼𝑈𝑛≥𝜇𝑛−𝛿𝜇𝑛+ğ¸ğ‘Ž+𝑋𝑛−𝛼𝐼𝑈𝑛<𝜇𝑛−𝛿𝜇𝑛≐𝑄1+𝑄2.(2.10) For 𝑄1, since 𝑋𝑛≥𝑈𝑛, we have 𝑄1î€·â‰¤ğ¸ğ‘Ž+ğ‘‹ğ‘›î€¸âˆ’ğ›¼ğ¼î€·ğ‘‹ğ‘›â‰¥ğœ‡ğ‘›âˆ’ğ›¿ğœ‡ğ‘›î€¸â‰¤î€·ğ‘Ž+𝜇𝑛−𝛿𝜇𝑛−𝛼.(2.11) By (2.8), we have for 𝑛≥𝑛0 that 𝜇𝑛−𝐸𝑈𝑛=𝐵𝑛𝑛−1𝑘=1𝐸𝑍𝑘𝐼𝑍𝑘>𝑏𝐵𝑛≤𝛿𝜇𝑛2.(2.12) Therefore, by (2.12), Markov’s inequality, Remark 2.2 and 𝐶𝑟’s inequality, for any 𝑝>2 and all 𝑛 sufficiently large, 𝑄2â‰¤ğ‘Žâˆ’ğ›¼ğ‘ƒî€·ğ‘ˆğ‘›<𝜇𝑛−𝛿𝜇𝑛=ğ‘Žâˆ’ğ›¼ğ‘ƒî€·ğ¸ğ‘ˆğ‘›âˆ’ğ‘ˆğ‘›>ğ›¿ğœ‡ğ‘›âˆ’î€·ğœ‡ğ‘›âˆ’ğ¸ğ‘ˆğ‘›î€¸î€¸â‰¤ğ‘Žâˆ’ğ›¼ğ‘ƒî‚µğ¸ğ‘ˆğ‘›âˆ’ğ‘ˆğ‘›>𝛿𝜇𝑛2î‚¶â‰¤ğ‘Žâˆ’ğ›¼ğ‘ƒî‚µ||𝑈𝑛−𝐸𝑈𝑛||>𝛿𝜇𝑛2≤𝐶𝜇𝑛−𝑝𝐸||𝑈𝑛−𝐸𝑈𝑛||𝑝≤𝐶𝜇𝑛−𝑝𝐵𝑛−2𝑛𝐸𝑍21𝐼𝑍1≤𝑏𝐵𝑛𝑝/2+𝐶𝜇𝑛−𝑝𝐵𝑛−𝑝𝑛𝐸𝑍𝑝1𝐼𝑍1≤𝑏𝐵𝑛≤𝐶𝜇𝑛−𝑝𝐵𝑛−1𝑛𝐸𝑍1𝐼𝑍1≤𝑏𝐵𝑛𝑝/2+𝐶𝜇𝑛−𝑝𝐵𝑛−1𝑛𝐸𝑍1𝐼𝑍1≤𝑏𝐵𝑛≤𝐶𝜇𝑛−𝑝𝜇𝑛𝑝/2+𝜇𝑛𝜇=𝐶𝑛−𝑝/2+𝜇𝑛−(𝑝−1).(2.13) Taking 𝑝>max{2,2𝛼,𝛼+1}, we have by (2.10), (2.11), and (2.13) that limsupğ‘›â†’âˆžî€·ğ‘Ž+ğœ‡ğ‘›î€¸ğ›¼ğ¸î€·ğ‘Ž+𝑋𝑛−𝛼≤limsupğ‘›â†’âˆžî€·ğ‘Ž+ğœ‡ğ‘›î€¸ğ›¼î€·ğ‘Ž+𝜇𝑛−𝛿𝜇𝑛−𝛼+limsupğ‘›â†’âˆžî€·ğ‘Ž+𝜇𝑛𝛼𝐶𝜇𝑛−𝑝/2+𝐶𝜇𝑛−(𝑝−1)=(1−𝛿)−𝛼,(2.14) which implies (2.7). This completes the proof of the theorem.

Proof of Corollary 1.3. The condition 𝐵𝑛=𝑂(𝑛𝛿) for some 0<𝛿<1 implies that 𝜇𝑛≐𝐸𝑋𝑛=𝐵𝑛𝑛−1𝑘=1𝐸𝑍𝑘=𝑛𝐵𝑛−1𝐸𝑍1,(2.15) thus, 𝜇𝑛≥𝐶𝑛1âˆ’ğ›¿â†’âˆž as ğ‘›â†’âˆž.
The fact 0<𝐸𝑍1<∞ and ğµğ‘›â†’âˆž yield that 𝐸𝑍1𝐼(𝑍1>𝑏𝐵𝑛)→0 as ğ‘›â†’âˆž, which implies that for all 0<𝜀<1, there exists 𝑛0>0 such that 𝐸𝑍1𝐼𝑍1>𝑏𝐵𝑛≤𝜀𝐸𝑍1,𝑛≥𝑛0.(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 Var𝑋𝑛. By Remark 2.2, 0<𝐸𝑍21<∞ and the condition 𝐵𝑛≥𝐶𝑛1/2 for all 𝑛 large enough, we can get that Var𝑋𝑛=𝐵𝑛−2Var𝑛𝑖=1𝑍𝑖≤𝐵𝑛−2𝐸𝑛𝑖=1𝑍𝑖2≤𝐶𝑛𝐵𝑛−2𝐸𝑍21≤𝐶1(2.17) for all 𝑛 large enough.
Denote 𝜙(𝑥)=(ğ‘Ž+𝑥)−𝛼 for 𝑥≥0. By Taylor’s expansion, we can see that 𝜙𝑋𝑛=𝜙𝐸𝑋𝑛+ğœ™î…žî€·ğœ‰ğ‘›ğ‘‹î€¸î€·ğ‘›âˆ’ğ¸ğ‘‹ğ‘›î€¸,(2.18) where 𝜉𝑛 is between 𝑋𝑛 and 𝐸𝑋𝑛. It is easily seen that {ğœ™î…ž(𝑥)}2 is decreasing in 𝑥≥0. Therefore, by (2.18), Cauchy-Schwartz inequality, (2.17) and (1.4), we have 𝑋𝐸𝜙𝑛−𝜙𝐸𝑋𝑛2𝜙=ğ¸î…žî€·ğœ‰ğ‘›ğ‘‹î€¸î€·ğ‘›âˆ’ğ¸ğ‘‹ğ‘›î€¸î€»2î€ºğœ™â‰¤ğ¸î…žî€·ğœ‰ğ‘›î€¸î€»2Var𝑋𝑛≤𝐶1ğ¸î€ºğœ™î…žî€·ğœ‰ğ‘›î€¸î€»2=𝐶1ğ¸î€ºğœ™î…žî€·ğœ‰ğ‘›î€¸î€»2𝐼𝑋𝑛≤𝐸𝑋𝑛+𝐶1ğ¸î€ºğœ™î…žî€·ğœ‰ğ‘›î€¸î€»2𝐼𝑋𝑛>𝐸𝑋𝑛≤𝐶1ğ¸î€ºğœ™î…žî€·ğ‘‹ğ‘›î€¸î€»2+𝐶1ğ¸î€ºğœ™î…žî€·ğ¸ğ‘‹ğ‘›î€¸î€»2∼2𝐶1ğ¸î€ºğœ™î…žî€·ğ¸ğ‘‹ğ‘›î€¸î€»2=2𝐶1𝛼2î€·ğ‘Ž+𝐸𝑋𝑛−2(𝛼+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 Var𝑋𝑛≤𝐶1, we make the use of Var𝑋𝑛≤𝐶𝑛𝐵𝑛−2𝐸𝑍21≐𝐶2𝑛𝐵𝑛−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).