# A Note on the Inverse Moments for Nonnegativeρ-Mixing Random Variables

Authors: Aiting Shen
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## 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 ofρ-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.


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## Body

## 1. Introduction

Firstly, we will recall the definition ofρ-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 ℱnm=σ(Xi,n≤i≤m) and ℱ𝒮=σ(Xi,i∈S⊂ℕ). Given σ-algebras ℬ,ℛ in ℱ, let
(1.1)ρ(B,R)=sup⁡X∈L2(B),Y∈L2(R)|EXY-EXEY|Var⁡(X)⋅Var⁡(Y).
Define the ρ-mixing coefficients by (1.2)ρ(n)=sup⁡k≥1ρ(F1k,Fk+n∞),n≥0.Definition 1.1.
A sequence{Xn,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 nonnegativeρ-mixing random variables with identical distribution.Let{Zn,n≥1} be a sequence of independent nonnegative random variables with finite second moments. Denote (1.3)Xn=∑i=1nZiBn,Bn2=∑i=1nVar⁡Zi.
It is interesting to show that under suitable conditions the following equivalence relation holds, namely,
(1.4)E(a+Xn)-r~(a+EXn)-r,n⟶∞,
where a>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 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)
EXn→∞, EZn3<∞;(iii)
(Lc condition) ∑i=1nE|Zi-EZi|c/Bnc→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 ∑n=1∞ρ(n)<∞.  Suppose that(i)
EZn2<∞, for all n≥1;(ii)
EXn→∞, where Xn is defined by (1.3);(iii)
for someη>0,(1.5)Rn(η)∶=Bn-2∑i=1nEZi2I(Zi>ηBn)⟶0,n⟶∞;(iv)
for somet∈(0,1) and any positive constants a,r,C,(1.6)lim⁡n→∞(a+EXn)r⋅exp⁡{-C⋅(EXn)tn}=0.
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 nonnegativeρ-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 a>0 and α>0 be real numbers. p>max{2,2α,α+1}.  Assume that ∑n=1∞ρ2/p(2n)<∞.  Suppose that(i)
0<EZn<∞, for all n≥1;(ii)
μn≐EXn→∞ as n→∞, where Xn=Bn-1∑k=1nZk;(iii)
for all0<ε<1, there exist b>0 and n0>0 such that(1.7)EZ1I(Z1>bBn)≤εEZ1,n≥n0.
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 Bn→∞ as n→∞. Let a>0 and α>0 be real numbers. p>max{2,2α,α+1}.  Assume that ∑n=1∞ρ2/p(2n)<∞. Then (1.4) holds.By Theorem1.2, we can get the following convergence rate of relative error in the relation (1.4).Theorem 1.4.
Assume that conditions of Theorem1.2 are satisfied and 0<EZn2<∞. p>max{2,4(α+1),2α+3}. If Bn≥Cn1/2 for all n large enough, where C is a positive constant, then
(1.8)|(a+EXn)αE(a+Xn)-α-1|=O((a+EXn)-1).Theorem 1.5.
Assume that conditions of Theorem1.2 are satisfied and 0<EZn2<∞. p>max{2,4(α+1),2α+3}.  Then
(1.9)|(a+EXn)αE(a+Xn)-α-1|=O(n-1/2).TakingBn≡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 a>0 and α>0 be real numbers. p>max{2,2α,α+1}.  Assume that ∑n=1∞ρ2/p(2n)<∞.  Suppose that(i)
0<EZn<∞, ∀n≥1;(ii)
νn≐EYn→∞ as n→∞, where Yn=∑k=1nZk;(iii)
for all0<ε<1, there exist b>0 and n0>0 such that
(1.10)EZ1I(Z1>b)≤εEZ1,n≥n0. Then E(a+Yn)-α~(a+EYn)-α.Remark 1.7.
Theorem1.2 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 EZn2<∞, 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 Bn2=∑i=1nVarZi 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
(1.11)Bn-1∑i=1nEZiI(Zi>ηBn)≤η-1Bn-2∑i=1nEZi2I(Zi>ηBn)⟶0,n⟶∞,
which implies that for all 0<ε<1, there exists a positive integer n0 such that
(1.12)Bn-1∑i=1nEZiI(Zi>ηBn)≤εμn=εBn-1∑i=1nEZi,n≥n0,
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 forρ-mixing random variables.Lemma 2.1 (c.f. Shao [18, Corollary  1.1]).
Letq≥2 and {Xn,n≥1} be a sequence of ρ-mixing random variables. Assume that EXn=0, E|Xn|q<∞ and
(2.1)∑n=1∞ρ2/q(2n)<∞.
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,
(2.2)E(max⁡1≤i≤n|Sk(i)|q)≤K[(nmax⁡k<i≤k+nEXi2)q/2+nmax⁡k<i≤k+nE|Xi|q],
where Sk(i)=∑j=k+1k+iXj, 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
(2.3)E(max⁡1≤i≤n|Sk(i)|q)≤K[(nEX12)q/2+nE|X1|q],E(max⁡1≤i≤n|∑j=1iXj|q)≤K[(nEX12)q/2+nE|X1|q]=K[(∑j=1nEXj2)q/2+∑j=1nE|Xj|q].
The inequality above is the Rosenthal type inequality of identical distributed ρ-mixing random variables.Proof of Theorem1.2.
It is easily seen thatf(x)=(a+x)-α is a convex function of x on [0,∞), therefore, we have by Jensen’s inequality that
(2.4)E(a+Xn)-α≥(a+EXn)-α,
which implies that
(2.5)lim⁡inf⁡n→∞(a+EXn)αE(a+Xn)-α≥1.
To prove (1.4), it is enough to prove that
(2.6)limsup⁡n→∞(a+EXn)αE(a+Xn)-α≤1.
In order to prove (2.6), we need only to show that for all δ∈(0,1),
(2.7)limsup⁡n→∞(a+EXn)αE(a+Xn)-α≤(1-δ)-α.
By (iii), we can see that for all δ∈(0,1),
(2.8)EZ1I(Z1>bBn)≤δ2EZ1,n≥n0.
Let
(2.9)Un=Bn-1∑k=1nZkI(Zk≤bBn),(2.10)E(a+Xn)-α=E(a+Xn)-αI(Un≥μn-δμn)+E(a+Xn)-αI(Un<μn-δμn)≐Q1+Q2.
For Q1, since Xn≥Un, we have
(2.11)Q1≤E(a+Xn)-αI(Xn≥μn-δμn)≤(a+μn-δμn)-α.
By (2.8), we have for n≥n0 that
(2.12)μn-EUn=Bn-1∑k=1nEZkI(Zk>bBn)≤δμn2.
Therefore, by (2.12), Markov’s inequality, Remark 2.2 and Cr’s inequality, for any p>2 and all n sufficiently large,
(2.13)Q2≤a-αP(Un<μn-δμn)=a-αP(EUn-Un>δμn-(μn-EUn))≤a-αP(EUn-Un>δμn2)≤a-αP(|Un-EUn|>δμn2)≤Cμn-pE|Un-EUn|p≤Cμn-p[Bn-2nEZ12I(Z1≤bBn)]p/2+Cμn-p[Bn-pnEZ1pI(Z1≤bBn)]≤Cμn-p[Bn-1nEZ1I(Z1≤bBn)]p/2+Cμn-pBn-1nEZ1I(Z1≤bBn)≤Cμn-p(μnp/2+μn)=C(μn-p/2+μn-(p-1)).
Taking p>max{2,2α,α+1}, we have by (2.10), (2.11), and (2.13) that
(2.14)limsup⁡n→∞(a+μn)αE(a+Xn)-α≤limsup⁡n→∞(a+μn)α(a+μn-δμn)-α+limsup⁡n→∞(a+μn)α[Cμn-p/2+Cμn-(p-1)]=(1-δ)-α,
which implies (2.7). This completes the proof of the theorem.Proof of Corollary1.3.
The conditionBn=O(nδ) for some 0<δ<1 implies that
(2.15)μn≐EXn=Bn-1∑k=1nEZk=nBn-1EZ1,
thus, μn≥Cn1-δ→∞ as n→∞.
The fact0<EZ1<∞ and Bn→∞ yield that EZ1I(Z1>bBn)→0 as n→∞, which implies that for all 0<ε<1, there exists n0>0 such that
(2.16)EZ1I(Z1>bBn)≤εEZ1,n≥n0.
That is to say condition (iii) of Theorem 1.2 holds. Therefore, the desired result follows from Theorem 1.2 immediately.Proof of Theorem1.4.
Firstly, we will examineVarXn. By Remark 2.2, 0<EZ12<∞ and the condition Bn≥Cn1/2 for all n large enough, we can get that
(2.17)Var⁡Xn=Bn-2Var⁡(∑i=1nZi)≤Bn-2E(∑i=1nZi)2≤CnBn-2EZ12≤C1
for all n large enough.
Denoteϕ(x)=(a+x)-α for x≥0. By Taylor’s expansion, we can see that
(2.18)ϕ(Xn)=ϕ(EXn)+ϕ′(ξn)(Xn-EXn),
where ξn is between Xn and EXn. 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
(2.19)[Eϕ(Xn)-ϕ(EXn)]2=E[ϕ′(ξn)(Xn-EXn)]2≤E[ϕ′(ξn)]2Var⁡Xn≤C1E[ϕ′(ξn)]2=C1E[ϕ′(ξn)]2I(Xn≤EXn)+C1E[ϕ′(ξn)]2I(Xn>EXn)≤C1E[ϕ′(Xn)]2+C1E[ϕ′(EXn)]2~2C1E[ϕ′(EXn)]2=2C1α2(a+EXn)-2(α+1).
This leads to (1.8). The proof is complete.Proof of Theorem1.5.
The proof is similar to that of Theorem1.4. In place of VarXn≤C1, we make the use of VarXn≤CnBn-2EZ12≐C2nBn-2. The proof is complete.



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