A Note on the Inverse Moments for Nonnegative 𝜌-Mixing Random Variables

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 {𝑋_𝑛,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 ℱ^𝑚𝑛=𝜎(𝑋_𝑖,𝑛≤𝑖≤𝑚) and ℱ_𝒮=𝜎(𝑋_𝑖,𝑖∈𝑆⊂ℕ). Given 𝜎-algebras ℬ,ℛ in ℱ, let 𝜌(ℬ,ℛ)=sup_{𝑋∈𝐿2(ℬ),𝑌∈𝐿2(ℛ)}||𝐸𝑋𝑌−𝐸𝑋𝐸𝑌||√Var(𝑋)⋅Var(𝑌).(1.1) Define the 𝜌-mixing coefficients by 𝜌(𝑛)=sup_𝑘≥1𝜌ℱ^𝑘1,ℱ^∞𝑘+𝑛,𝑛≥0.(1.2)

Definition 1.1. A sequence {𝑋_𝑛,𝑛≥1} of random variables is said to be 𝜌-mixing if 𝜌(𝑛)↓0 as 𝑛→∞.

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 {𝑍_𝑛,𝑛≥1} be a sequence of independent nonnegative random variables with finite second moments. Denote 𝑋_𝑛=∑^𝑛𝑖=1𝑍_𝑖𝐵_𝑛,𝐵^2𝑛=^𝑛_𝑖=1Var𝑍_𝑖.(1.3) It is interesting to show that under suitable conditions the following equivalence relation holds, namely, 𝐸𝑎+𝑋_𝑛^−𝑟∼𝑎+𝐸𝑋_𝑛^−𝑟,𝑛⟶∞,(1.4) where 𝑎>0 and 𝑟>0 are arbitrary real numbers.

Here and below, for two positive sequences {𝑐_𝑛,𝑛≥1} and {𝑑_𝑛,𝑛≥1}, we write 𝑐_𝑛∼𝑑_𝑛 if 𝑐_𝑛𝑑^𝑛−1→1 as 𝑛→∞. 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)𝐸𝑋_𝑛→∞, 𝐸𝑍^3𝑛<∞;(iii)(𝐿_𝑐 condition) ∑^𝑛𝑖=1𝐸|𝑍_𝑖−𝐸𝑍_𝑖|^𝑐/𝐵^𝑐𝑛→0 (𝑐=3).

Hu et al. [12] considered weaker conditions: 𝐸𝑍^𝑛2+𝛿<∞, where 𝑍_𝑛 satisfies 𝐿_2+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 𝐵_𝑛=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 {𝑍_𝑛,𝑛≥1} be a nonnegative 𝜌-mixing sequence with ∑^∞𝑛=1𝜌(𝑛)<∞. Suppose that (i)𝐸𝑍^2𝑛<∞, for all 𝑛≥1;(ii)𝐸𝑋_𝑛→∞, where 𝑋_𝑛 is defined by (1.3);(iii)for some 𝜂>0,𝑅_𝑛(𝜂)∶=𝐵^𝑛𝑛−2_𝑖=1𝐸𝑍^2𝑖𝐼𝑍_𝑖>𝜂𝐵_𝑛⟶0,𝑛⟶∞;(1.5)(iv)for some 𝑡∈(0,1) and any positive constants 𝑎,𝑟,𝐶,lim_𝑛→∞𝑎+𝐸𝑋_𝑛^𝑟⋅exp−𝐶⋅𝐸𝑋^𝑛^𝑡𝑛=0.(1.6) Then for any 𝑎>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 {𝑍_𝑛,𝑛≥1} be a sequence of nonnegative 𝜌-mixing random variables with identical distribution and let {𝐵_𝑛,𝑛≥1} be a sequence of positive constants. Let 𝑎>0 and 𝛼>0 be real numbers. 𝑝>max{2,2𝛼,𝛼+1}. Assume that ∑^∞𝑛=1𝜌^2/𝑝(2^𝑛)<∞. Suppose that (i)0<𝐸𝑍_𝑛<∞, for all 𝑛≥1;(ii)𝜇_𝑛≐𝐸𝑋_𝑛→∞ as 𝑛→∞, where 𝑋_𝑛=𝐵^𝑛−1∑^𝑛𝑘=1𝑍_𝑘;(iii)for all 0<𝜀<1, there exist 𝑏>0 and 𝑛₀>0 such that 𝐸𝑍₁𝐼𝑍₁>𝑏𝐵_𝑛≤𝜀𝐸𝑍₁,𝑛≥𝑛₀.(1.7) Then (1.4) holds.

Corollary 1.3. Let {𝑍_𝑛,𝑛≥1} be a sequence of nonnegative 𝜌-mixing random variables with identical distribution and 0<𝐸𝑍₁<∞. Let {𝐵_𝑛,𝑛≥1} be a sequence of positive constants satisfying 𝐵_𝑛=𝑂(𝑛^𝛿) for some 0<𝛿<1 and 𝐵_𝑛→∞ as 𝑛→∞. Let 𝑎>0 and 𝛼>0 be real numbers. 𝑝>max{2,2𝛼,𝛼+1}. Assume that ∑^∞𝑛=1𝜌^2/𝑝(2^𝑛)<∞. 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<𝐸𝑍^2𝑛<∞. 𝑝>max{2,4(𝛼+1),2𝛼+3}. If 𝐵_𝑛≥𝐶𝑛^1/2 for all 𝑛 large enough, where 𝐶 is a positive constant, then ||𝑎+𝐸𝑋_𝑛^𝛼𝐸𝑎+𝑋_𝑛^−𝛼−1||=𝑂𝑎+𝐸𝑋_𝑛⁻¹.(1.8)

Theorem 1.5. Assume that conditions of Theorem 1.2 are satisfied and 0<𝐸𝑍^2𝑛<∞. 𝑝>max{2,4(𝛼+1),2𝛼+3}. Then ||𝑎+𝐸𝑋_𝑛^𝛼𝐸𝑎+𝑋_𝑛^−𝛼−1||=𝑂𝑛^−1/2.(1.9)

Taking 𝐵_𝑛≡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 {𝑍_𝑛,𝑛≥1} be a sequence of nonnegative 𝜌-mixing random variables with identical distribution. Let 𝑎>0 and 𝛼>0 be real numbers. 𝑝>max{2,2𝛼,𝛼+1}. Assume that ∑^∞𝑛=1𝜌^2/𝑝(2^𝑛)<∞. Suppose that (i)0<𝐸𝑍_𝑛<∞, ∀𝑛≥1;(ii)𝜈_𝑛≐𝐸𝑌_𝑛→∞ as 𝑛→∞, where 𝑌_𝑛=∑^𝑛𝑘=1𝑍_𝑘;(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(𝑎+𝑌_𝑛)^−𝛼∼(𝑎+𝐸𝑌_𝑛)^−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 𝐸𝑍_𝑛<∞, for all 𝑛≥1, which is weaker than the condition 𝐸𝑍^2𝑛<∞, for all 𝑛≥1 in the above cited references. Secondly, {𝐵_𝑛,𝑛≥1} is an arbitrary sequence of positive constants in Theorem 1.2, while 𝐵^2𝑛=∑^𝑛𝑖=1Var𝑍_𝑖 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 𝐵^𝑛𝑛−1_𝑖=1𝐸𝑍_𝑖𝐼𝑍_𝑖>𝜂𝐵_𝑛≤𝜂⁻¹𝐵^𝑛𝑛−2_𝑖=1𝐸𝑍^2𝑖𝐼𝑍_𝑖>𝜂𝐵_𝑛⟶0,𝑛⟶∞,(1.11) which implies that for all 0<𝜀<1, there exists a positive integer 𝑛₀ such that 𝐵^𝑛𝑛−1_𝑖=1𝐸𝑍_𝑖𝐼𝑍_𝑖>𝜂𝐵_𝑛≤𝜀𝜇_𝑛=𝜀𝐵^𝑛𝑛−1_𝑖=1𝐸𝑍_𝑖,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 𝑞≥2 and {𝑋_𝑛,𝑛≥1} be a sequence of 𝜌-mixing random variables. Assume that 𝐸𝑋_𝑛=0, 𝐸|𝑋_𝑛|^𝑞<∞ and ^∞_𝑛=1𝜌^2/𝑞(2^𝑛)<∞.(2.1) Then there exists a positive constant 𝐾=𝐾(𝑞,𝜌(⋅)) depending only on 𝑞 and 𝜌(⋅) such that for any 𝑘≥0 and 𝑛≥1, 𝐸max_{1≤𝑖≤𝑛}||𝑆_𝑘(𝑖)||^𝑞≤𝐾𝑛max_{𝑘<𝑖≤𝑘+𝑛}𝐸𝑋^2𝑖^𝑞/2+𝑛max_{𝑘<𝑖≤𝑘+𝑛}𝐸||𝑋_𝑖||^𝑞,(2.2) where 𝑆_𝑘(𝑖)=∑^{𝑘+𝑖𝑗=𝑘+1}𝑋_𝑗, 𝑘≥0 and 𝑖≥1.

Remark 2.2. We point out that if {𝑋_𝑛,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max_{1≤𝑖≤𝑛}||𝑆_𝑘(𝑖)||^𝑞≤𝐾𝑛𝐸𝑋²¹^𝑞/2𝐸⎛⎜⎜⎝max_{1≤𝑖≤𝑛}+𝑛𝐸||𝑋₁||^𝑞,|||||_𝑖_𝑗=1𝑋_𝑗|||||^𝑞⎞⎟⎟⎠≤𝐾𝑛𝐸𝑋₂₁=𝐾⎡⎢⎢⎣_𝑛^𝑞/2+𝑛𝐸||𝑋₁||^𝑞_𝑗=1𝐸𝑋^2𝑗^𝑞/2+^𝑛_𝑗=1𝐸||𝑋_𝑗||^𝑞⎤⎥⎥⎦.(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)=(𝑎+𝑥)^−𝛼 is a convex function of 𝑥 on [0,∞), therefore, we have by Jensen’s inequality that 𝐸𝑎+𝑋_𝑛^−𝛼≥𝑎+𝐸𝑋_𝑛^−𝛼,(2.4) which implies that liminf_𝑛→∞𝑎+𝐸𝑋_𝑛^𝛼𝐸𝑎+𝑋_𝑛^−𝛼≥1.(2.5) To prove (1.4), it is enough to prove that limsup_𝑛→∞𝑎+𝐸𝑋_𝑛^𝛼𝐸𝑎+𝑋_𝑛^−𝛼≤1.(2.6) In order to prove (2.6), we need only to show that for all 𝛿∈(0,1), limsup_𝑛→∞𝑎+𝐸𝑋_𝑛^𝛼𝐸𝑎+𝑋_𝑛^−𝛼≤(1−𝛿)^−𝛼.(2.7) By (iii), we can see that for all 𝛿∈(0,1), 𝐸𝑍₁𝐼𝑍₁>𝑏𝐵_𝑛≤𝛿2𝐸𝑍¹,𝑛≥𝑛₀.(2.8) Let 𝑈_𝑛=𝐵^𝑛𝑛−1_𝑘=1𝑍_𝑘𝐼𝑍_𝑘≤𝑏𝐵_𝑛,(2.9)𝐸𝑎+𝑋_𝑛^−𝛼=≐𝐸𝑄₁𝑎+𝑋_𝑛^−𝛼𝐼𝑈_𝑛≥𝜇_𝑛−𝛿𝜇_𝑛+𝐸𝑎+𝑋_𝑛^−𝛼𝐼𝑈_𝑛<𝜇_𝑛−𝛿𝜇_𝑛+𝑄₂.(2.10) For 𝑄₁, since 𝑋_𝑛≥𝑈_𝑛, we have 𝑄₁≤𝐸𝑎+𝑋_𝑛^−𝛼𝐼𝑋_𝑛≥𝜇_𝑛−𝛿𝜇_𝑛≤𝑎+𝜇_𝑛−𝛿𝜇_𝑛^−𝛼.(2.11) By (2.8), we have for 𝑛≥𝑛₀ that 𝜇_𝑛−𝐸𝑈_𝑛=𝐵^𝑛𝑛−1_𝑘=1𝐸𝑍_𝑘𝐼𝑍_𝑘>𝑏𝐵_𝑛≤𝛿𝜇^𝑛2.(2.12) Therefore, by (2.12), Markov’s inequality, Remark 2.2 and 𝐶_𝑟’s inequality, for any 𝑝>2 and all 𝑛 sufficiently large, 𝑄₂=≤𝑎𝑎^−𝛼_−𝛼𝑃𝑈_𝑛<𝜇_𝑛−𝛿𝜇_𝑛≤𝑎_−𝛼𝑃𝐸𝑈_𝑛−𝑈_𝑛>𝛿𝜇_𝑛−𝜇_𝑛−𝐸𝑈_𝑛𝑃𝐸𝑈_𝑛−𝑈_𝑛>𝛿𝜇^𝑛≤𝑎_−𝛼2𝑃||𝑈_𝑛−𝐸𝑈_𝑛||>𝛿𝜇^𝑛2≤𝐶𝜇^𝑛−𝑝𝐸||𝑈_𝑛−𝐸𝑈_𝑛||^𝑝≤≤𝐶𝐶𝜇𝜇^𝑛−𝑝_𝑛−𝑝𝐵^𝑛−2𝑛𝐸𝑍²¹𝐼𝑍₁≤𝑏𝐵_𝑛^𝑝/2+𝐶𝜇^𝑛−𝑝𝐵^𝑛−𝑝𝑛𝐸𝑍^𝑝1𝐼𝑍₁≤𝑏𝐵_𝑛≤𝐶𝜇_𝑛−𝑝𝐵^𝑛−1𝑛𝐸𝑍₁𝐼𝑍₁≤𝑏𝐵_𝑛^𝑝/2+𝐶𝜇^𝑛−𝑝𝐵^𝑛−1𝑛𝐸𝑍₁𝐼𝑍₁≤𝑏𝐵_𝑛𝜇^𝑛𝑝/2+𝜇_𝑛=𝐶𝜇^{𝑛−𝑝/2}+𝜇^{𝑛−(𝑝−1)}.(2.13) Taking 𝑝>max{2,2𝛼,𝛼+1}, we have by (2.10), (2.11), and (2.13) that limsup_𝑛→∞𝑎+𝜇_𝑛^𝛼𝐸𝑎+𝑋_𝑛^−𝛼=≤(li1m−su𝛿p_𝑛→∞)_−𝛼𝑎+𝜇_𝑛^𝛼𝑎+𝜇_𝑛−𝛿𝜇_𝑛^−𝛼+limsup_𝑛→∞𝑎+𝜇_𝑛^𝛼𝐶𝜇^{𝑛−𝑝/2}+𝐶𝜇^{𝑛−(𝑝−1)},(2.14) which implies (2.7). This completes the proof of the theorem.

Proof of Corollary 1.3. The condition 𝐵_𝑛=𝑂(𝑛^𝛿) for some 0<𝛿<1 implies that 𝜇_𝑛≐𝐸𝑋_𝑛=𝐵^𝑛𝑛−1_𝑘=1𝐸𝑍_𝑘=𝑛𝐵^𝑛−1𝐸𝑍₁,(2.15) thus, 𝜇_𝑛≥𝐶𝑛^1−𝛿→∞ as 𝑛→∞.
The fact 0<𝐸𝑍₁<∞ and 𝐵_𝑛→∞ yield that 𝐸𝑍₁𝐼(𝑍₁>𝑏𝐵_𝑛)→0 as 𝑛→∞, which implies that for all 0<𝜀<1, there exists 𝑛₀>0 such that 𝐸𝑍₁𝐼𝑍₁>𝑏𝐵_𝑛≤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 Var𝑋_𝑛. By Remark 2.2, 0<𝐸𝑍²¹<∞ and the condition 𝐵_𝑛≥𝐶𝑛^1/2 for all 𝑛 large enough, we can get that Var𝑋_𝑛=𝐵^𝑛−2Var^𝑛_𝑖=1𝑍_𝑖≤𝐵^𝑛−2𝐸^𝑛_𝑖=1𝑍_𝑖²≤𝐶𝑛𝐵^𝑛−2𝐸𝑍²¹≤𝐶₁(2.17) for all 𝑛 large enough.
Denote 𝜙(𝑥)=(𝑎+𝑥)^−𝛼 for 𝑥≥0. By Taylor’s expansion, we can see that 𝜙𝑋_𝑛=𝜙𝐸𝑋_𝑛+𝜙^𝜉_𝑛𝑋_𝑛−𝐸𝑋_𝑛,(2.18) where 𝜉_𝑛 is between 𝑋_𝑛 and 𝐸𝑋_𝑛. It is easily seen that {𝜙^(𝑥)}² is decreasing in 𝑥≥0. Therefore, by (2.18), Cauchy-Schwartz inequality, (2.17) and (1.4), we have 𝐸𝜙𝑋_𝑛−𝜙𝐸𝑋_𝑛²=𝐸𝜙^𝜉_𝑛𝑋_𝑛−𝐸𝑋_𝑛²≤𝐸𝜙^𝜉_𝑛²Var𝑋_𝑛≤𝐶₁𝐸𝜙^𝜉_𝑛²=≤𝐶𝐶₁₁𝐸𝜙^𝜉_𝑛²𝐼𝑋_𝑛≤𝐸𝑋_𝑛+𝐶₁𝐸𝜙^𝜉_𝑛²𝐼𝑋_𝑛>𝐸𝑋_𝑛𝐸𝜙^𝑋_𝑛²+𝐶₁𝐸𝜙^𝐸𝑋_𝑛²∼2𝐶₁𝐸𝜙^𝐸𝑋_𝑛²=2𝐶₁𝛼²𝑎+𝐸𝑋_𝑛^−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 Var𝑋_𝑛≤𝐶₁, we make the use of Var𝑋_𝑛≤𝐶𝑛𝐵^𝑛−2𝐸𝑍²¹≐𝐶₂𝑛𝐵^𝑛−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).