Sufficient and necessary conditions of convergence for ρ͠ mixing random variables

Shui-Li Zhang; Yu Miao; Cong Qu

doi:10.1515/math-2019-0036

Artikel Open Access

Sufficient and necessary conditions of convergence for ρ͠ mixing random variables

Shui-Li Zhang , Yu Miao und Cong Qu

Veröffentlicht/Copyright: 30. Mai 2019

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Abstract

In the present paper, the sufficient and necessary conditions of the complete convergence and complete moment convergence for ρ͠-mixing random variables are established, which extend some well-known results.

Keywords: Complete convergence; complete moment convergence; ρ͠-mixing random variables; weakly bounded

MSC 2010: 60F15

1 Introduction

1.1 ρ͠-mixing sequence

Let (Ω, 𝓕, ℙ) be a probability space, {X_n, n ≥ 1} be a sequence of random variables defined on (Ω, 𝓕, ℙ), a Sn=∑i=1nXi for n ≥ 1. For any S ⊂ N = {1, 2, ⋯}, define 𝓕_S = σ(X_i, i ∈ S). Let 𝓐 and 𝓑 be two sub σ-algebra on 𝓕, put

ρ(A,B)=sup|EXY−EXEY|E(X−EX)2E(Y−EY)2:X∈L2(A),Y∈L2(B). (1.1)

Define the ρ͠-mixing coefficients by

ρ~n=sup{ρ(FS,FT):S,T⊂N with dist(S,T)≥n}, (1.2)

where dist(S, T) = inf{|s − t| : s ∈ S, t ∈ T}. Obviously, 0 ≤ ρ͠_n+1 ≤ ρ͠_n ≤ ρ͠₀ = 1. Then the sequence {X_n, n ≥ 1} is called ρ͠-mixing if there exists k ∈ N such that ρ͠_k < 1.

The notion of ρ͠-mixing random variables was first introduced by Bradley [1], and a number of limits results for ρ͠-mixing random variables have been established by many authors. One can refer to Bradley [1] for the central limit theorem; Sung [2, 3], An and Yuan [4], Lan [5], Guo and Zhu [6] for complete convergence; Zhang [7] for complete moment convergence; Peligrad and Gut [8], Utev and Peligrad [9] for the moment inequalities; Gan [10], Wu and Jiang [11], Kuczmaszewska [12] for strong law of large numbers.

1.2 Some notations and known results

Let {X_n, n ≥ 1} be a sequence of random variables and if there exist a positive constant C₁ (C₂) and a random variable X, such that the left-hand side (right-hand side) of the following inequalities is satisfied for all n ≥ 1 and x ≥ 0,

C1P(|X|>x)≤1n∑k=1nP(|Xk|>x)≤C2P(|X|>x), (1.3)

then the sequence {X_n, n ≥ 1} is said to be weakly lower (upper) bounded by X. The sequence {X_n, n ≥ 1} is said to be weakly bounded by X if it is both weakly lower and upper bounded by X. A sequence of random variables {U_n, n ≥ 1} is said to converge completely to a constant C if

∑n=1∞P|Un−C|>ε<∞,for allε>0. (1.4)

The concept of complete convergence was introduced firstly by Hsu and Robbins [13]. In view of the Borel-Cantelli lemma, complete convergence implies that U_n → C almost surely.

The complete moment convergence is a more general concept than the complete convergence, which was introduced by Chow [14]. Let {Z_n, n ≥ 1} be a sequence of random variables and a_n > 0, b_n > 0, q > 0, if

∑n=1∞anE{bn−1|Zn|−ϵ}+q<∞, for some or all ϵ>0,

then the above result was called the complete moment convergence. The complete convergence and complete moment convergence have been studied by many authors. For instance, see Wang [15], Zhao [16], Zhang [17] and so on.

Baum and Katz [18] obtained the following equivalent conditions for the i.i.d. random variables.

Theorem A

(Baum and Katz [18]) Let 0 < r < 2, r ≤ q p. Suppose that {X_n, n ≥ 1} is a sequence of i.i.d. random variables with mean zero, then 𝔼 |X₁|^p < ∞ is equivalent to the condition that

∑n=1∞np/r−2P|Sn|>εn1/r<∞, for all ε>0, (1.5)

and also equivalent to the condition that

∑n=1∞np/r−2Pmax1≤k≤n|Sk|>εn1/r<∞, for all ε>0. (1.6)

For the i.i.d. case, related results are fruitful and detailed. It is natural to extend them to dependent case, for examples, martingale difference, negatively associated, mixing random variables and so on. In the present paper, we are interested in the ρ͠-mixing random variables.

For identically distributed ρ͠-mixing random variables, Peligrad and Gut [8] extended the results of Baum and Katz to ρ͠-mixing random variables (see Theorem B); subsequently, An and Yuan [4] extended the results of Peligrad and Gut [8] to weighted sums of ρ͠-mixing random variables (see Theorem C); Gan [10] obtained a sufficient condition on complete convergence (see Theorem D).

Theorem B

(Peligrad and Gut [8]) Let {X_n, n ≥ 1} be a sequence of identically distributed ρ͠-mixing random variables, α p > 1, α > 1/2, and suppose that 𝔼 X₁ = 0 for α ≤ 1. Assume that lim_n→∞ρ͠_n < 1, then 𝔼 |X₁|^p < ∞ is equivalent to the condition that

∑n=1∞nαp−2Pmax1≤j≤n|Sj|>εnα<∞, for all ε>0. (1.7)

Theorem C

(An and Yuan [4]) Let {X_n, n ≥ 1} be a sequence of identically distributed ρ͠-mixing random variables, α p > 1, α > 1/2 and suppose that 𝔼 X₁ = 0 for α ≤ 1. Assume that {a_ni, 1 ≤ i ≤ n} is an array of real numbers satisfying

∑i=1n|ani|p=O(nδ),0<δ<1

and

♯Ank=♯{1≤i≤n:|ani|p>(k+1)−1}≥ne−1/k.

Then 𝔼 |X₁|^p < ∞ is equivalent to

∑n=1∞nαp−2Pmax1≤j≤n∑i=1janiXi>εnα<∞, for all ε>0. (1.8)

Theorem D

(Gan [10]) Let {X_n, n ≥ 1} be a sequence of identically distributed ρ͠-mixing random variables with ρ͠(1) < 1 and 1 < p ≤ 2, δ > 0, α ≥ max{(1 + δ)/p, 1}. If 𝔼 X₁ = 0 and 𝔼 |X₁|^p < ∞, then

∑n=1∞nαp−2−δP|Sn|>εnα<∞, for all ε>0. (1.9)

In this paper, the purpose is to study and establish the equivalent conditions on complete convergence and complete moment convergence for ρ͠-mixing random variables. Our main results are stated in Section 2 and all proofs are given in Section 3. Throughout the paper, C denotes a positive constant not depending on n, which may be different in various places. Let I(A) be the indicator function of the set A, a_n = O(b_n) represent a_n ≤ Cb_n for all n ≥ 1.

2 Main results

In the section, we state our main results and some remarks. Recall that a real-valued function l(x), positive and measurable on (0, ∞), is said to be slowly varying at infinity if limx→∞l(λx)/l(x)=1 for each λ > 0. Let

L={f:f(x) is slowly varying function, and such that∫1kxsf(x)dx≥Cks+1f(k), for all k>1,s>−1}.

Firstly, we state the complete convergence for the weighted sums of {X_n, n ≥ 1}.

Theorem 2.1

Let {X_n, n ≥ 1} be a sequence of ρ͠-mixing random variables which is weakly bounded by X. Let α > 1/2, α p ≥ 1 and 𝔼 X_n = 0 if p > 1. Assume that l(x) ∈ 𝓛 and {a_ni, 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying max1≤i≤n|ani|=O(n−α) . Then 𝔼|X|^pl(|X|^1/α) < ∞ is equivalent to

∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXi>ε<∞, for all ε>0. (2.1)

Remark 2.1

When right-hand side of inequality (1.3) is satisfied and l(x) is slowly varying function, the moment condition 𝔼|X|^pl(|X|^1/α) < ∞ still implied (2.1). Conversely, for the sufficient condition of Theorem 2.1, we need left-hand side of inequality (1.3) and l ∈ 𝓛.

Since log x ∈ 𝓛 and 1 ∈ 𝓛, we can obtain the following corollaries.

Corollary 2.1

Let {X_n, n ≥ 1} be a sequence of ρ͠-mixing random variables which is weakly bounded by X. Let α > 1/2, α p ≥ 1 and 𝔼 X_n = 0 if p > 1. Assume that {a_ni, 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying max1≤i≤n|ani|=O(n−α) . Then 𝔼|X|^plog(|X|) < ∞ is equivalent to

∑n=1∞nαp−2log⁡nPmax1≤k≤n∑i=1kaniXi>ε<∞, for all ε>0. (2.2)

Corollary 2.2

∑n=1∞nαp−2Pmax1≤k≤n∑i=1kaniXi>ε<∞, for all ε>0. (2.3)

Remark 2.2

Let {X_n, n ≥ 1} be a sequence of identically distributed ρ͠-mixing random variables, then {X_n, n ≥ 1} is weakly bounded by X₁. By taking a_ni = n^−α for all 1 ≤ i ≤ n, n ≥ 1, then Theorem B can be obtained by Corollary 2.2. Moreover, we not only consider the case α p > 1, but also consider the case α p = 1, so Theorem 2.1 extended and improved well-known results.

Remark 2.3

For independent random variable sequence, we have ρ͠_n = 0 for all n ≥ 1. So our results extend the Baum-Katz theorem from i.i.d. case to non-identically distributed ρ͠-mixing random variables.

Next, we give the complete moment convergence for the weighted sums of {X_n, n ≥ 1}.

Theorem 2.2

Let {X_n, n ≥ 1} be a sequence of ρ͠-mixing random variables which is weakly bounded by X. Let p > 1, α > 1/2, α p ≥ 1 and 𝔼 X_n = 0. Assume that l(x) ∈ 𝓛 and {a_ni, 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying max1≤i≤n|ani|=O(n−α) . Then 𝔼|X|^pl(|X|^1/α) < ∞ is equivalent to

∑n=1∞nαp−2l(n)Emax1≤k≤n∑i=1kaniXi−ϵ+<∞, for all ε>0. (2.4)

Remark 2.4

Similar to Theorem 2.1, the necessary condition of Theorem 2.2 only need that the right-hand side of inequality (1.3) and l(x) is slowly varying function, the sufficient condition of Theorem 2.2 need the left-hand side of inequality (1.3) and l ∈ 𝓛.

Corollary 2.3

Let {X_n, n ≥ 1} be a sequence of ρ͠-mixing random variables which is weakly bounded by X. Let p > 1, α > 1/2, α p ≥ 1 and 𝔼 X_n = 0. Assume {a_ni, 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying max1≤i≤n|ani|=O(n−α) . Then 𝔼|X|^plog(|X|) < ∞ is equivalent to

∑n=1∞nαp−2log⁡(n)Emax1≤k≤n∑i=1kaniXi−ϵ+<∞, for all ε>0. (2.5)

Corollary 2.4

∑n=1∞nαp−2Emax1≤k≤n∑i=1kaniXi−ϵ+<∞, for all ε>0. (2.6)

3 Proofs of Main results

3.1 Some lemmas

To prove our results, we first give some lemmas as follows.

Lemma 3.1

(Kuzmaszewska [19]) Let {X_n, n ≥ 1} be a sequence of random variables which is weakly mean dominated (or weakly upper bounded) by a random variable X. If 𝔼|X|^p < ∞ for some p > 0, then for any t > 0 and n ≥ 1, the following statements hold:

1n∑k=1nE|Xk|p≤CE|X|p, (3.1)

1n∑k=1nE|Xk|pI(|Xk|≤t)≤CE|X|pI(|X|≤t)+tpP(|X|>t) (3.2)

and

1n∑k=1nE|Xk|pI(|Xk|>t)≤CE|X|pI(|X|>t). (3.3)

Lemma 3.2

(Lan [5]) Let {X_n, n ≥ 1} be a sequence of ρ͠-mixing random variables, then there exists a positive constant C such that for any x ≥ 0 and all n ≥ 1,

12−Pmax1≤k≤n|Xk|>x∑k=1nP(|Xk|>x)≤C2+1Pmax1≤k≤n|Xk|>x. (3.4)

Lemma 3.3

(Sung [20]) Let {Y_n, n ≥ 1} and {Z_n, n ≥ 1} be sequences of random variables, then for any q > 1, ϵ > 0 and all a > 0, we have

Emax1≤j≤n∑i=1j(Yi+Zi)−ϵa+≤1ϵq+1q−11aq−1Emax1≤j≤n∑i=1jYiq+Emax1≤j≤n∑i=1jZi. (3.5)

Lemma 3.4

(Utev and Peligrad [9]) For a positive integer N ≥ 1 and positive real numbers q ≥ 2 and 0 ≤ r < 1, there is a positive constant C = C(q, N, r) such that if {X_n, n ≥ 1} is a sequence of random variables with ρ͠_N ≤ r, with 𝔼 X_k = 0 and 𝔼 |X_k|^q < ∞ for every k ≥ 1, then for all n ≥ 1,

Emax1≤j≤n∑i=1jXiq≤C∑i=1nE|Xi|q+∑i=1nEXi2q/2. (3.6)

Lemma 3.5

(Zhou [21]) Let l(x) be slowly varying at infinity, then we have

(i)∑k=1nkpl(k)≤Cnp+1l(n), for p>−1 and positive integer n;

(ii) ∑k=n∞kpl(k)≤Cnp+1l(n), for p<−1 and positive integer n.

Lemma 3.6

(Bai [22]) Let l(x) be slowly varying at infinity, then we have

limx→∞l(kx)l(x)=1, for any k>0; limx→∞l(x+u)l(x)=1, for any u>0;
limx→∞xαl(x)=∞; limx→∞x−αl(x)=0, for any α>0;
limn→∞sup2n≤x<2n+1l(x)l(2n)=1;
C12nrl(ϵ2n)≤∑k=1n2krl(ϵ2k)≤C22nrl(ϵ2n) for every r > 0, ϵ > 0 and positive integer n.

3.2 Proof of Theorem 2.1

Without loss of generality, we can assume that a_ni > 0 for all 1 ≤ i ≤ n, n ≥ 1. For fixed n, let

Xni=XiI(|Xi|≤nα), Xni′=XiI(|Xi|>nα), i≥1.

We will consider the following three cases, p > 1, p = 1 and 0 < p < 1 respectively.

Let p > 1. It is easy to see that

∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXi>ε≤∑n=1∞nαp−2l(n)P⋃i=1n(|Xi|>nα) +∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXni>ε≜I+J. (3.7)

In order to prove (2.1), it need only to show that I < ∞ and J < ∞. From (1.3), Lemma 3.5 and Markov’s inequality, we have

I=∑n=1∞nαp−2l(n)P⋃i=1n(|Xi|>nα)≤∑n=1∞nαp−2l(n)∑i=1nP(|Xi|>nα)≤C∑n=1∞nαp−1l(n)P(|X|>nα)≤C∑n=1∞nαp−1−αl(n)E[|X|I(|X|>nα)]≤C∑n=1∞nαp−α−1l(n)∑k=n∞E[|X|I(kα≤|X|<(k+1)α)]≤C∑k=1∞E[|X|I(kα≤|X|<(k+1)α)]∑n=1knαp−α−1l(n)≤C∑k=1∞E[|X|I(kα≤|X|<(k+1)α)]kαp−αl(k)≤C∑k=1∞E[|X|pl(|X|1/α)I(kα≤|X|<(k+1)α)]≤CE[|X|pl(|X|1/α)]<∞. (3.8)

Note that α p ≥ 1, 𝔼 X_n = 0, by Lemma 3.1, then

max1≤j≤n∑i=1janiEXni≤max1≤j≤n∑i=1janiEXiI(|Xi|>nα)≤∑i=1naniE|Xi|I(|Xi|>nα)≤C1nα∑i=1nE|Xi|I(|Xi|>nα)≤C1nαp−1E|X|pI(|X|>nα)→0. (3.9)

In order to prove J < ∞, from (3.9), it is enough to check

∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kani(Xni−EXni)>ε2<∞.

By taking q>max{p,2,αpα−12} and from Lemma 3.4, we have that

∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kani(Xni−EXni)>ε2≤C∑n=1∞nαp−2l(n)Emax1≤k≤n∑i=1kani(Xni−EXni)q≤C∑n=1∞nαp−2l(n)∑i=1naniqE|Xni−EXni|q+∑i=1nani2E(Xni−EXni)2q/2≜J1+J2. (3.10)

Note that q > p, by (3.8) and Lemma 3.1, Lemma 3.5, then

J1≤C∑n=1∞nαp−2l(n)∑i=1naniqE|Xni|q≤C∑n=1∞nαp−2−αql(n)∑i=1nE|Xi|qI(|Xi|≤nα)≤C∑n=1∞nαp−1−αql(n)E|X|qI(|X|≤nα)+nαqP(|X|>nα)≤C∑n=1∞nαp−1−αql(n)∑k=1nE|X|qI((k−1)α<|X|≤kα)+C∑n=1∞nαp−1l(n)P(|X|>nα)≤C∑k=1∞E|X|qI((k−1)α<|X|≤kα)∑n=k∞nαp−1−αql(n)≤C∑k=1∞E|X|qI((k−1)α<|X|≤kα)kαp−αql(k)≤CE|X|pl(|X|1/α)<∞. (3.11)

In order to get J₂ < ∞, we consider the following cases.
1. p ≥ 2, α p > 1. From q>(αp−1)/(α−12) and Lemma 3.6, we can get that
  
  J2=∑n=1∞nαp−2l(n)∑i=1nani2E(Xni−EXni)2q/2≤C∑n=1∞nαp−2−αql(n)∑i=1nEXni2q/2≤C∑n=1∞nαp−2−αqnq/2l(n)EX2q/2≤C∑n=1∞nαp−2−αq+q/2l(n)<∞. (3.12)
2. 1 < p < 2, α p > 1. Take q = 2, by Lemma 3.1 and (3.8), then
  
  J2=∑n=1∞nαp−2l(n)∑i=1nani2E(Xni−EXni)2≤C∑n=1∞nαp−2−2αl(n)∑i=1nEXi2I(|Xi|≤nα)≤C∑n=1∞nαp−1−2αl(n)[EX2I(|X|≤nα)+n2αP(|X|>nα)]≤C∑n=1∞nαp−1−2αl(n)∑k=1nEX2I((k−1)α<|X|≤kα)+∑n=1∞nαp−1l(n)P(|X|>nα)]≤C∑k=1∞EX2I((k−1)α<|X|≤kα)∑n=k∞nαp−1−2αl(n)≤C∑k=1∞EX2I((k−1)α<|X|≤kα)kαp−2αl(k)≤CE|X|pl(|X|1/α)<∞. (3.13)
3. p > 1, α p = 1, α > 12 . Take q = 2, similar to the proof of (3.13), it following that J₂ < ∞. From the above discussions, we can get (2.1) for the case p > 1.
Let p = 1. By the similar proof of (3.9), we have

max1≤j≤n∑i=1janiEXni→0. (3.14)

So in order to get (2.1), it is enough to show

∑n=1∞nα−2l(n)P⋃i=1n(|Xi|>nα)<∞ (3.15)

and

∑n=1∞nα−2l(n)Pmax1≤k≤n∑i=1kaniXni−∑i=1kaniEXni>ε2<∞. (3.16)

By the condition (1.3) and Lemma 3.5, we have

∑n=1∞nα−2l(n)P⋃i=1n(|Xi|>nα)≤∑n=1∞nα−2l(n)∑i=1nP(|Xi|>nα)≤C∑n=1∞nα−1l(n)P(|X|>nα)≤C∑n=1∞nα−1l(n)∑k=n∞P(kα<|X|≤(k+1)α)≤C∑k=1∞P(kα<|X|≤(k+1)α)∑n=1knα−1l(n)≤C∑k=1∞P(kα<|X|≤(k+1)α)kαl(k)≤C∑k=1∞E[|X|l(|X|1/α)I(kα<|X|≤(k+1)α)]≤CE[|X|l(|X|1/α)]<∞. (3.17)

Furthermore, from (3.17), Lemma 3.4 and Lemma 3.5, we get

∑n=1∞nα−2l(n)Pmax1≤k≤n∑i=1kaniXni−∑i=1kaniEXni>ε2≤C∑n=1∞nα−2l(n)Emax1≤k≤n∑i=1kani(Xni−EXni)2≤C∑n=1∞n−α−2l(n)∑i=1nE|Xni−EXni|2≤C∑n=1∞n−α−1l(n)E|X|2I(|X|≤nα)+n2αP(|X|>nα)≤C∑n=1∞n−α−1l(n)∑k=1nE|X|2I((k−1)α<|X|≤kα)≤C∑k=1∞E|X|2I((k−1)α<|X|≤kα)∑n=k∞n−α−1l(n)≤C∑k=1∞E|X|2I((k−1)α<|X|≤kα)k−αl(k)≤CE|X|l(|X|1/α)<∞. (3.18)

Based on the above discussions, we can get (2.1) for the case p = 1.
Let 0 < p < 1. Since X_i = X_ni + Xni′ for all i ≥ 1, we have

∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXi>ε≤∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXni>ε2+∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXni′>ε2≜I∗+J∗. (3.19)

By Lemma 3.1 and Lemma 3.5, we obtain that

I∗=∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXni>ε2≤C∑n=1∞nαp−α−2l(n)∑i=1nEXni≤C∑n=1∞nαp−α−1l(n)E|X|I(|X|≤nα)+nαP(|X|>nα)≤C∑n=1∞nαp−α−1l(n)∑k=1nE|X|I((k−1)α<|X|≤kα) +C∑n=1∞nαp/2−1l(n)∑k=n∞E|X|p/2I(kα≤|X|<(k+1)α)≤C∑k=1∞E|X|I((k−1)α<|X|≤kα)∑n=k∞nαp−α−1l(n) +C∑k=1∞E|X|p/2I(kα≤|X|<(k+1)α)∑n=1knαp/2−1l(n)≤C∑k=1∞E|X|I((k−1)α<|X|≤kα)kαp−αl(k)+C∑k=1∞E|X|p/2I(kα≤|X|<(k+1)α)kαp/2l(k)≤CE|X|pl(|X|1/α)<∞. (3.20)

and

J∗=∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXni′>ε2 ≤C∑n=1∞nαp/2−2l(n)E∑i=1n|Xni′|p/2 ≤C∑n=1∞nαp/2−2l(n)∑i=1nE|Xni′|p/2 ≤C∑n=1∞nαp/2−1l(n)E|X|p/2I(|X|>nα) ≤C∑n=1∞nαp/2−1l(n)∑k=n∞E|X|p/2I(kα≤|X|<(k+1)α) ≤C∑k=1∞E|X|p/2I(kα≤|X|<(k+1)α)kαp/2l(k) ≤CE|X|pl(|X|1/α)<∞. (3.21)

From (3.19)-(3.21), we can see that (2.1) holds for 0 < p < 1.

Conversely, we take a_ni = n^−α for all 1 ≤ i ≤ n, n ≥ 1, then

∑n=1∞nαp−2l(n)Pmax1≤i≤n|Xi|>ϵnα≤C∑n=1∞nαp−2l(n)P2max1≤j≤n∑i=1jXi>ϵnα≤C∑n=1∞nαp−2l(n)Pmax1≤j≤n∑i=1janiXi>ϵ2<∞. (3.22)

Because of α p ≥ 1, we have

limn→∞Pmax1≤i≤n|Xi|>ϵnα→0. (3.23)

Thus, for all n large enough, it follows that

Pmax1≤i≤n|Xi|>ϵnα<14. (3.24)

Using Lemma 3.2 and (3.24), we get

nC1P(|X|>ϵnα)≤∑k=1nP(|Xk|>ϵnα)≤(2C+4)Pmax1≤k≤n|Xk|>ϵnα. (3.25)

Hence by l ∈ 𝓛, we obtain

∞>∑n=1∞nαp−2l(n)Pmax1≤i≤n|Xi|>nα≥C∑n=1∞nαp−2l(n)nP(|X|>nα)=C∑n=1∞nαp−1l(n)∑k=n∞P(kα<|X|≤(k+1)α)≥C∑k=1∞P(kα<|X|≤(k+1)α)∑n=1knαp−1l(n)≥C∑k=1∞P(kα<|X|≤(k+1)α)kαpl(k)≥CE|X|pl(|X|1/α), (3.26)

which implies the desired results.

3.3 Proof of Theorem 2.2

First we assume that the claim (2.4) holds, then it follows that

∞>∑n=1∞nαp−2l(n)Emax1≤k≤n∑i=1kaniXi−ϵ+=∑n=1∞nαp−2l(n)∫0∞Pmax1≤k≤n∑i=1kaniXi−ϵ>tdt≥∑n=1∞nαp−2l(n)∫0ϵPmax1≤k≤n∑i=1kaniXi>ϵ+tdt≥C∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXi>2ϵ. (3.27)

From Theorem 2.1, we know that 𝔼 |X|^pl(|X|^1/α) < ∞ is equivalent to

∑n=1∞nαp−2l(n)Pmax1≤k≤n∑i=1kaniXi>ϵ<∞.

Conversely, if 𝔼 |X|^pl(|X|^1/α) < ∞, then from Lemma 3.3, we have that

∑n=1∞nαp−2l(n)Emax1≤k≤n∑i=1kaniXi−ϵ+≤∑n=1∞nαp−2l(n)Emax1≤k≤n∑i=1kani[(Xni−EXni)+(Xni′−EXni′)]−ϵ+≤C∑n=1∞nαp−2l(n)Emax1≤k≤n∑i=1kani(Xni−EXni)q+∑n=1∞nαp−2l(n)Emax1≤k≤n∑i=1kani(Xni′−EXni′)≜I~+J~. (3.28)

From (3.10)-(3.13), we have that I͠ < ∞. Furthermore, by (3.8) and Lemma 3.1, we obtain

J~≤C∑n=1∞nαp−2−αl(n)Emax1≤k≤n∑i=1k(Xni′−EXni′)≤C∑n=1∞nαp−2−αl(n)∑i=1nE|Xi|I(|Xi|>nα)≤C∑n=1∞nαp−1−αl(n)E|X|I(|X|>nα)<∞. (3.29)

From (3.28) and (3.29), we can get (2.4).

Acknowledgement

This work is supported by IRTSTHN (14IRTSTHN023), NSFC (11471104).

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Received: 2018-10-05

Accepted: 2019-01-31

Published Online: 2019-05-30

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Artikel in diesem Heft

https://doi.org/10.1515/math-2019-0036

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Complete convergence; complete moment convergence; ρ͠-mixing random variables; weakly bounded

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