Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums of PNQD Random Variables

Abstract

The complete convergence for pairwise negative quadrant dependent (PNQD) random variables is studied. So far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. We establish a collection that contains relationship to overcome the difficulties that there is no general moment inequality. Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by Baum and Katz (1965) and Wu (2002).

1. Introduction and Lemmas

Random variables 𝑋 and 𝑌 are said to be negative quadrant dependent (NQD) if 𝑃𝑋≤𝑥,𝑌≤𝑦≤𝑃(𝑋≤𝑥)𝑃𝑌≤𝑦, (1.1) for all 𝑥,𝑦∈𝑅. A sequence of random variables {𝑋_𝑛;𝑛≥1} is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the sequence is NQD. This definition was introduced by Lehmann [1]. Obviously, PNQD sequence includes many negatively associated sequences, and pairwise independent random sequence is the most special case.

In many mathematics and mechanic models, a PNQD assumption among the random variables in the models is more reasonable than an independence assumption. PNQD series have received more and more attention recently because of their wide applications in mathematics and mechanic models, percolation theory, and reliability theory. Many statisticians have investigated PNQD series with interest and have established a series of useful results. For example, Matula [2], Li and Yang [3], and Wu and Jiang [4] obtained the strong law of large numbers, Wang et al. [5] obtained the Marcinkiewicz's weak law of large numbers, Wu [6] obtained the strong convergence properties of Jamison weighted sums, the three-series theorem, and complete convergence theorem, and Li and Wang [7] obtained the central limit theorem. It is interesting for us to extend the limit theorems to the case of PNQD series. However, so far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. In the above-mentioned conclusions, only the Kolmogorov-type strong law of large numbers obtained by Matula [2, Theorem 1] and Baum and Katz-type complete convergence theorem obtained by Wu [6, Theorem 4] achieve the corresponding conclusions of independent cases, and the rest did not achieve the optimal results of independent cases.

Complete convergence is one of the most important problems in probability theory. Recent results of the complete convergence can be found in Wu [6], Chen and Wang [8], and Li et al. [9]. In this paper, we establish a collection that contains relationship to overcome the difficulties that there is no the general moment inequality and obtain the complete convergence theorem for weighted sums of PNQD sequence, which extend and improve the corresponding results of Baum and Katz [10] and Wu [6].

Lemma 1.1 (see [1]). Let 𝑋 and 𝑌 be NQD random variables. Then (i)cov(𝑋,𝑌)≤0, (ii)𝑃(𝑋>𝑥,𝑌>𝑦)≤𝑃(𝑋>𝑥)𝑃(𝑌>𝑦), for all 𝑥,𝑦∈R, (iii)if 𝑓 and 𝑔 are Borel functions, both of which being monotone increasing (or both are monotone decreasing), then 𝑓(𝑋) and 𝑔(𝑌) are NQD.

Lemma 1.2 (see [6, Lemma 2]). Let {𝑋_𝑛;𝑛≥1} be a sequence of PNQD random variables with 𝐸𝑋_𝑛=0,𝐸𝑋²_𝑛<∞,𝑇_𝑗(𝑘)=∑^𝑗+𝑘_𝑖=𝑗+1𝑋_𝑖,𝑗≥0. Then 𝐸𝑇_𝑗(𝑘)²≤^𝑗+𝑘_𝑖=𝑗+1𝐸𝑋²_𝑖,𝐸max_{1≤𝑘≤𝑛}𝑇_𝑗(𝑘)²≤4log²𝑛log²2^𝑗+𝑛_𝑖=𝑗+1𝐸𝑋²_𝑖. (1.2)

Lemma 1.3 (see [2, Lemma 1]). (i) If ∑^∞_𝑛=1𝑃(𝐴_𝑛)<∞, then 𝑃(𝐴_𝑛;𝑖.𝑜.)=0.
(ii) if 𝑃(𝐴_𝑘𝐴_𝑚)≤𝑃(𝐴_𝑘)𝑃(𝐴_𝑚),𝑘≠𝑚, and ∑^∞_𝑛=1𝑃(𝐴_𝑛)=∞, then 𝑃(𝐴_𝑛;𝑖.𝑜.)=1.

Lemma 1.4. Let {𝑋_𝑛;𝑛≥1} be a sequence of PNQD random variables. Then for any 𝑥≥0, there exists a positive constant 𝑐 such that for all 𝑛≥1, 1−𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘>𝑥^2𝑛_𝑘=1𝑃𝑋_𝑘>𝑥≤𝑐𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘>𝑥. (1.3)

Proof. We can prove the Lemma by Lemma A.6 of Zhang and Wen [11].

2. Main Results and the Proof

In the following, the symbol 𝑐 stands for a generic positive constant which may differ from one place to another. Let 𝑎_𝑛≪𝑏_𝑛 (𝑎_𝑛≫𝑏_𝑛) denote that there exists a constant 𝑐>0 such that 𝑎_𝑛≤𝑐𝑏_𝑛 (𝑎_𝑛≥𝑐𝑏_𝑛) for all sufficiently large 𝑛, and let 𝑋_𝑖≺𝑋 (𝑋_𝑖≻𝑋) denote that there exists a constant 𝑐>0 such that 𝑃(|𝑋_𝑖|>𝑥)≤𝑐𝑃(|𝑋|>𝑥) (𝑃(|𝑋_𝑖|>𝑥)≥𝑐𝑃(|𝑋|>𝑥)) for all 𝑖≥1 and 𝑥>0.

Theorem 2.1. Let {𝑋_𝑛;𝑛≥1} be a sequence of PNQD random variables with 𝑋_𝑖≺𝑋. Let {𝑎_𝑛𝑘;𝑘≤𝑛,𝑛≥1} be a sequence of real numbers such that 𝑎_𝑛𝑘≪𝑛^−𝛼,∀𝑘≤𝑛,𝑛≥1. (2.1) Let for 𝛼𝑝>1,0<𝑝<2,𝛼>0, and 𝐸𝑋_𝑖=0, for 𝛼≤1. If 𝐸|𝑋|^𝑝<∞, (2.2)
then ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}𝑆_𝑛𝑘>𝜀<∞,∀𝜀>0, (2.3) where 𝑆_𝑛𝑘=∑^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖.

Theorem 2.2. Let {𝑋_𝑛;𝑛≥1} be a sequence of PNQD random variables with 𝑋_𝑖≻𝑋. Let {𝑎_𝑛𝑘;𝑘≤𝑛,𝑛≥1} be a sequence of real numbers such that |𝑎_𝑛𝑘|≫𝑛^−𝛼, for all 𝑘≤𝑛,𝑛≥1. Let for 𝛼>0,𝛼𝑝>1,0<𝑝<2. If (2.3) holds, then (2.2) holds.

Remark 2.3. Taking 𝑎_𝑛𝑖=𝑛^−𝛼, for all 𝑖≤𝑛,𝑛≥1 in Theorem 2.1, then ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}𝑆_𝑛𝑘>𝜀=^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}𝑛^−𝛼𝑘_𝑖=1𝑋_𝑖=>𝜀_∞𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑖>𝜀𝑛^𝛼. (2.4) Hence, Theorem 4 in Wu [6] is a particular case of our Theorem 2.1.

Remark 2.4. When {𝑋_𝑛;𝑛≥1} is i.i.d. and 𝑎_𝑛𝑖=𝑛^−𝛼, for all 𝑖≤𝑛,𝑛≥1, then Theorems 2.1 and 2.2 become Baum and Katz [10] complete convergence theorem. Hence, our Theorems 2.1 and 2.2 improve and extend the well-known Baum and Katz theorem.

Proof of Theorem 2.1. Without loss of generality, assume that 𝑎_𝑛𝑘>0 for 𝑘≤𝑛,𝑛≥1. Let 𝑞>0 such that (1+(1/𝛼𝑝))/2<𝑞<1. For all 𝑖≤𝑛, let 𝑌_𝑛𝑖=−𝑎⁻¹_𝑛𝑖𝑛^{𝛼(𝑞−1)}𝐼𝑎_𝑛𝑖𝑋_𝑖<−𝑛^{𝛼(𝑞−1)}+𝑋_𝑖𝐼𝑎_𝑛𝑖𝑋_𝑖≤𝑛^{𝛼(𝑞−1)}+𝑎⁻¹_𝑛𝑖𝑛^{𝛼(𝑞−1)}𝐼𝑎_𝑛𝑖𝑋_𝑖>𝑛^{𝛼(𝑞−1)},𝑈_𝑛𝑘=^𝑘_𝑖=1𝑎_𝑛𝑖𝑌_𝑛𝑖. (2.5) Write 𝐴_𝑛=^𝑛_𝑗=1𝑎_𝑛𝑗𝑋_𝑗,𝐵≥𝜀_𝑛=_{1≤𝑖<𝑗≤𝑛}𝑎_𝑛𝑖𝑋_𝑖>𝑛^{𝛼(𝑞−1)},𝑎_𝑛𝑗𝑋_𝑗>𝑛^{𝛼(𝑞−1)}𝑎_𝑛𝑖𝑋_𝑖<−𝑛^{𝛼(𝑞−1)},𝑎_𝑛𝑗𝑋_𝑗<−𝑛^{𝛼(𝑞−1)}. (2.6)
Firstly, we prove that max_{1≤𝑘≤𝑛}𝑆_𝑛𝑘<6𝜀⊇𝐴^𝑐_𝑛max_{1≤𝑘≤𝑛}𝑈_𝑛𝑘<2𝜀𝐵^𝑐_𝑛=^𝑛_𝑗=1𝑎_𝑛𝑗𝑋_𝑗<𝜀max_{1≤𝑘≤𝑛}𝑈_𝑛𝑘<2𝜀_{1≤𝑖<𝑗≤𝑛}𝑎_𝑛𝑖𝑋_𝑖≤𝑛^{𝛼(𝑞−1)}𝑎_𝑛𝑗𝑋_𝑗≤𝑛^{𝛼(𝑞−1)}𝑎_𝑛𝑖𝑋_𝑖≥−𝑛^{𝛼(𝑞−1)}𝑎_𝑛𝑗𝑋_𝑗≥−𝑛^{𝛼(𝑞−1)}=𝐷_𝑛. (2.7)
For any 𝜔∈𝐷_𝑛, we have 𝑎_𝑛𝑗𝑋_𝑗<𝜀,𝑎_𝑛𝑗𝑌_𝑛𝑗≤𝑎_𝑛𝑗𝑋_𝑗<𝜀,∀1≤𝑗≤𝑛,max_{1≤𝑘≤𝑛}𝑈_𝑛𝑘<2𝜀, (2.8) and for any 1≤𝑖<𝑗≤𝑛, 𝑎_𝑛𝑖𝑋_𝑖≤𝑛^{𝛼(𝑞−1)},or𝑎_𝑛𝑗𝑋_𝑗≤𝑛^{𝛼(𝑞−1)},𝑎_𝑛𝑖𝑋_𝑖≥−𝑛^{𝛼(𝑞−1)},or𝑎_𝑛𝑗𝑋_𝑗≥−𝑛^{𝛼(𝑞−1)}. (2.9) Hence 𝑎=♯𝑖;1≤𝑖≤𝑛,𝑎_𝑛𝑖𝑋_𝑖(𝜔)>𝑛^{𝛼(𝑞−1)}≤1,𝑏=♯𝑖;1≤𝑖≤𝑛,𝑎_𝑛𝑖𝑋_𝑖(𝜔)<−𝑛^{𝛼(𝑞−1)}≤1, (2.10) where the symbol ♯𝐴 denotes the number of elements in the set 𝐴.
When 𝑎=𝑏=0, then |𝑎_𝑛𝑖𝑋_𝑖(𝜔)|≤𝑛^{𝛼(𝑞−1)} for any 1≤𝑖≤𝑛; thus, 𝑌_𝑛𝑖(𝜔)=𝑋_𝑖(𝜔), and therefore by (2.8), max_{1≤𝑘≤𝑛}𝑆_𝑛𝑘=max_{1≤𝑘≤𝑛}𝑈_𝑛𝑘<2𝜀<6𝜀. (2.11)
When 𝑎=1,𝑏=0 (or 𝑎=0,𝑏=1), then there exists only an 𝑖₀: 1≤𝑖₀≤𝑛 such that 𝑎_𝑛𝑖0𝑋_𝑖0(𝜔)>𝑛^{𝛼(𝑞−1)} (or 𝑎_𝑛𝑖0𝑋_𝑖0(𝜔)<−𝑛^{𝛼(𝑞−1)}), the remaining 𝑗,|𝑎_𝑛𝑗𝑋_𝑛𝑗(𝜔)|≤𝑛^{𝛼(𝑞−1)}; thus, 𝑋_𝑗(𝜔)=𝑌_𝑛𝑗(𝜔). If 1≤𝑘≤𝑖₀−1, then 𝑆_𝑛𝑘(𝜔)=𝑈_𝑛𝑘(𝜔). If 𝑖₀≤𝑘≤𝑛, then by (2.8), max_{1≤𝑘≤𝑛}𝑆_𝑛𝑘(𝜔)=max_{1≤𝑘≤𝑛1≤𝑖≤𝑘,𝑖≠𝑖0}𝑎_𝑛𝑖𝑋_𝑖(𝜔)+𝑎_𝑛𝑖0𝑋_𝑖0(𝜔)=max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑌_𝑛𝑖(𝜔)−𝑎_𝑛0𝑌_𝑛𝑖0(𝜔)+𝑎_𝑛𝑖0𝑋_𝑖0(𝜔)≤max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑌_𝑛𝑖(𝜔)+𝑎_𝑛𝑖0𝑌_𝑛𝑖0(𝜔)+𝑎_𝑛𝑖0𝑋_𝑖0(𝜔)<2𝜀+𝜀+𝜀<6𝜀. (2.12)
When 𝑎=𝑏=1, then there exist 1≤𝑖₁,𝑖₂≤𝑛 such that 𝑎_𝑛𝑖1𝑋_𝑖1(𝜔)>𝑛^{𝛼(𝑞−1)},𝑎_𝑛𝑖2𝑋_𝑖2(𝜔)<−𝑛^{𝛼(𝑞−1)}, the remaining 𝑗,|𝑎_𝑛𝑗𝑋_𝑗(𝜔)|≤𝑛^{𝛼(𝑞−1)}; thus, 𝑋_𝑗(𝜔)=𝑌_𝑛𝑗(𝜔). Without loss of generality, assume that 𝑖₁≤𝑖₂. If 1≤𝑘≤𝑖₁−1, then 𝑆_𝑛𝑘(𝜔)=𝑈_𝑛𝑘(𝜔); if 𝑖₁≤𝑘<𝑖₂, then by (2.8), max_{1≤𝑘≤𝑛}𝑆_𝑛𝑘(𝜔)≤max_{1≤𝑘≤𝑛}𝑈_𝑛𝑘(𝜔)+𝑎_𝑛𝑖1𝑌_𝑛𝑖1(𝜔)+𝑎_𝑛𝑖1𝑋_𝑖1(𝜔)<2𝜀+𝜀+𝜀<6𝜀. (2.13) If 𝑘≥𝑖₂, then by (2.8), max_{1≤𝑘≤𝑛}𝑆_𝑛𝑘(𝜔)=max_{1≤𝑘≤𝑛1≤𝑖≤𝑘,𝑖≠𝑖1,𝑖2}𝑎_𝑛𝑖𝑋_𝑖(𝜔)+𝑎_𝑛𝑖1𝑋_𝑖1(𝜔)+𝑎_𝑛𝑖2𝑋_𝑖2(𝜔)≤max_{1≤𝑘≤𝑛}𝑈_𝑛𝑘(𝜔)+𝑎_𝑛𝑖1𝑌_𝑛𝑖1(𝜔)+𝑎_𝑛𝑖2𝑌_𝑛𝑖2(𝜔+𝑎_𝑛𝑖1)𝑋_𝑖1(𝜔)+𝑎_𝑛𝑖2𝑋_𝑖2(𝜔)<6𝜀. (2.14) Hence, (2.7) holds, that is: max_{1≤𝑘≤𝑛}𝑆_𝑛𝑘≥6𝜀⊆𝐴_𝑛max_{1≤𝑘≤𝑛}𝑈_𝑛𝑘≥2𝜀𝐵_𝑛. (2.15) Therefore, in order to prove (2.3), we only need to prove that ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃𝐴_𝑛<∞,_∞𝑛=1𝑛^𝛼𝑝−2𝑃𝐵_𝑛<∞,_∞𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}𝑈_𝑛𝑘≥2𝜀<∞,∀𝜀>0. (2.16)(2.17)(2.18) By (2.1), (2.2), 𝑋_𝑖≺𝑋, and 𝛼𝑝>1, ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃𝐴_𝑛≤^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2}_𝑗=1𝑃𝑎_𝑛𝑗𝑋_𝑗≤≥𝜀_∞𝑛=1𝑛^{𝑛𝛼𝑝−2}_𝑗=1𝑃𝑋_𝑗≥𝜀𝑎⁻¹_𝑛𝑗≥𝜀𝑐𝑛^𝛼≪^∞_𝑛=1𝑛^𝛼𝑝−1𝑃|𝑋|≥𝜀𝑐𝑛^𝛼=^∞_𝑛=1𝑛^{∞𝛼𝑝−1}_𝑗=𝑛𝑃𝜀𝑐𝑗^𝛼≤|𝑋|<𝜀𝑐𝑗+1^𝛼=^∞𝑗_𝑗=1𝑛=1𝑛^𝛼𝑝−1𝑃𝜀𝑐𝑗^𝛼≤|𝑋|<𝜀𝑐𝑗+1^𝛼≤^∞_𝑗=1𝑗^𝛼𝑝𝑃𝜀𝑐𝑗^𝛼≤|𝑋|<𝜀𝑐𝑗+1^𝛼≪𝐸|𝑋|^𝑝<∞. (2.19) That is, (2.16) holds.
By Lemma 1.1(ii), 𝑋_𝑖≺𝑋, and the definition of 𝑞,𝛼𝑝(1−2𝑞)<−1, ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃𝐵_𝑛≤^∞_𝑛=1𝑛^𝛼𝑝−2_{1≤𝑖<𝑗≤𝑛}𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑛^{𝛼(𝑞−1)}𝑃𝑎_𝑛𝑗𝑋_𝑗>𝑛^{𝛼(𝑞−1)}+𝑃𝑎_𝑛𝑖𝑋_𝑖<−𝑛^{𝛼(𝑞−1)}𝑃𝑎_𝑛𝑗𝑋_𝑗<−𝑛^{𝛼(𝑞−1)}≪^∞_𝑛=1𝑛^𝛼𝑝𝑃²|𝑋|>𝑐𝑛^𝛼𝑞≤^∞_𝑛=1𝑛^𝛼𝑝𝑛^{−2𝛼𝑝𝑞}𝐸|𝑋|^𝑝2≪^∞_𝑛=1𝑛^{𝛼𝑝(1−2𝑞)}<∞. (2.20) That is, (2.17) holds.
In order to prove (2.18), firstly, we prove that max_{1≤𝑘≤𝑛}𝐸𝑈_𝑛𝑘=max_{1≤𝑘≤𝑛}𝐸^𝑘_𝑖=1𝑎_𝑛𝑖𝑌_𝑛𝑖⟶0,𝑛⟶∞. (2.21)
(i) When 𝛼≤1, then 𝑝>1/𝛼≥1; from 𝐸𝑋_𝑖=0 and the definition of 𝑞, we have 𝑞<1,𝛼𝑝𝑞>𝛼𝑝+1−𝛼𝑝𝑞=1+𝛼𝑝(1−𝑞)>1∶max_{1≤𝑘≤𝑛}𝐸^𝑘_𝑖=1𝑎_𝑛𝑖𝑌_𝑛𝑖≤^𝑛_𝑖=1𝑎_𝑛𝑖𝐸𝑌_𝑛𝑖=^𝑛_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖−𝑌_𝑛𝑖≤^𝑛_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖+𝑎⁻¹_𝑛𝑖𝑛^{𝛼(𝑞−1)}𝐼_{(𝑎𝑛𝑖𝑋𝑖<−𝑛𝛼(𝑞−1))}+𝐸𝑋_𝑖−𝑎⁻¹_𝑛𝑖𝑛^{𝛼(𝑞−1)}𝐼_{(𝑎𝑛𝑖𝑋𝑖>𝑛𝛼(𝑞−1))}≪^𝑛_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖𝐼_{(|𝑎𝑛𝑖𝑋𝑖|>𝑛𝛼(𝑞−1))}≤^𝑛_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖𝑎_𝑛𝑖𝑋_𝑖𝑛^{𝛼(𝑞−1)𝑝−1}=^𝑛_𝑖=1𝑎^𝑝_𝑛𝑖𝐸𝑋_𝑖^𝑝𝑛^{𝛼(1−𝑞)(𝑝−1)}≪𝑛^{−𝛼𝑝+1+𝛼𝑝−𝛼−𝛼𝑝𝑞+𝛼𝑞}=𝑛^{−(𝛼𝑝𝑞−1)−𝛼(1−𝑞)}⟶0,𝑛⟶∞. (2.22)
(ii) When 𝛼>1, and 𝑝≥1, then 𝐸|𝑋|<∞ from (2.2), thus, max_{1≤𝑘≤𝑛}𝐸^𝑘_𝑖=1𝑎_𝑛𝑖𝑌_𝑛𝑖≤^𝑛_𝑖=1𝑎_𝑛𝑖𝐸|𝑋|≪𝑛^−𝛼+1⟶0,𝑛⟶∞. (2.23)
(iii) When 𝛼>1, and 𝑝<1, by −(𝛼𝑝−1)−𝛼(1−𝑞)(1−𝑝)<0, and −𝛼(1−𝑞)−(𝛼𝑝𝑞−1)<0, we get max_{1≤𝑘≤𝑛}𝐸^𝑘_𝑖=1𝑎_𝑛𝑖𝑌_𝑛𝑖≤^𝑛_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑖𝐼_{(𝑎𝑛𝑖|𝑋𝑖|≤𝑛𝛼(𝑞−1))}+𝑎⁻¹_𝑛𝑖𝑛^{𝛼(𝑞−1)}𝑃𝑎_𝑛𝑖𝑋_𝑖>𝑛^{𝛼(𝑞−1)}≤^𝑛_𝑖=1𝑎^𝑝_𝑛𝑖𝐸𝑋_𝑖^𝑝𝑎_𝑛𝑖𝑋_𝑖^1−𝑝𝐼_{(𝑎𝑛𝑖|𝑋𝑖|≤𝑛𝛼(𝑞−1))}+^𝑛_𝑖=1𝑛^{𝛼(𝑞−1)}𝑎^𝑝_𝑛𝑖𝑛^{−𝛼𝑝(𝑞−1)}𝐸𝑋_𝑖^𝑝≪𝑛^{−(𝛼𝑝−1)−𝛼(1−𝑝)(1−𝑞)}+𝑛^{−𝛼(1−𝑞)−(𝛼𝑝𝑞−1)}⟶0. (2.24) Hence, (2.21) holds; that is, for any 𝜀>0, we have max_{1≤𝑘≤𝑛}|𝐸𝑈_𝑛𝑘|<𝜀 for all sufficiently large 𝑛. Thus, 𝑃max_{1≤𝑗≤𝑛}𝑈_𝑛𝑗≥2𝜀≤𝑃max_{1≤𝑗≤𝑛}𝑈_𝑛𝑗−𝐸𝑈_𝑛𝑗>𝜀. (2.25)
Let 𝑌_𝑛𝑖=𝑌_𝑛𝑖−𝐸𝑌_𝑛𝑖. Obviously, 𝑌_𝑛𝑖 is monotonic on 𝑋_𝑖. By Lemma 1.1(iii), {𝑌_𝑛𝑖;𝑛≥1,𝑖≤𝑛} is also a sequence of PNQD random variables with 𝐸𝑌_𝑛𝑖=0, by Lemma 1.2 and −1−𝛼(1−𝑞)(2−𝑝)<−1: ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑗≤𝑛}𝑈_𝑛𝑗−𝐸𝑈_𝑛𝑗≪>𝜀_∞𝑛=1𝑛^𝛼𝑝−2log²𝑛^𝑛_𝑗=1𝐸𝑎²_𝑛𝑗𝑌²_𝑛𝑗≪^∞_𝑛=1𝑛^𝛼𝑝−2log²𝑛^𝑛_𝑗=1𝐸𝑎²_𝑛𝑗𝑋²_𝑗𝐼_{(𝑎𝑛𝑗|𝑋𝑗|≤𝑛𝛼(𝑞−1))}+𝑛^{2𝛼(𝑞−1)}𝑃𝑎_𝑛𝑗𝑋_𝑗>𝑛^{𝛼(𝑞−1)}≤^∞_𝑛=1𝑛^𝛼𝑝−2log²𝑛^𝑛_𝑗=1𝐸𝑎_𝑛𝑗𝑋_𝑗^𝑝𝑛^{𝛼(𝑞−1)(2−𝑝)}+𝑛^{2𝛼(𝑞−1)−𝛼𝑝(𝑞−1)}𝐸𝑎_𝑛𝑗𝑋_𝑗^𝑝≪^∞_𝑛=1𝑛^{𝛼𝑝−1−𝛼𝑝+𝛼(𝑞−1)(2−𝑝)}+𝑛^{−1+𝛼𝑝−𝛼𝑝𝑞+2𝛼𝑞−2𝛼}log²𝑛=2^∞_𝑛=1𝑛^{−1−𝛼(1−𝑞)(2−𝑝)}log²𝑛<∞. (2.26) This completes the proof of Theorem 2.1.

Proof of Theorem 2.2. Noting that max_{1≤𝑘≤𝑛}|𝑎_𝑛𝑘𝑋_𝑘|≤2max_{1≤𝑘≤𝑛}|𝑆_𝑛𝑘| and |𝑎_𝑛𝑘|≫𝑛^−𝛼, from (2.3), ^∞_𝑛=1𝑛^𝛼𝑝−2𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘>𝜀𝑛^𝛼<∞,∀𝜀>0. (2.27) Thus, by 𝛼𝑝−2>−1, we get ^∞_𝑗=1𝑃max_{1≤𝑘≤2𝑗}𝑋_𝑘>𝜀2^𝛼(𝑗+1)≪^∞_𝑛=1𝑛⁻¹𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘>𝜀𝑛^𝛼<∞,∀𝜀>0. (2.28) This implies that max_{2𝑚−1≤𝑛≤2𝑚}𝑃max_{1≤𝑗≤𝑛}𝑋_𝑗>𝜀2^2𝛼𝑛^𝛼≤𝑃max_{1≤𝑗≤2𝑚}𝑋_𝑗>𝜀2^𝛼(𝑚+1)⟶0. (2.29) Hence, for all sufficiently large 𝑛, 𝑃max_{1≤𝑗≤𝑛}𝑋_𝑗>𝜀2^2𝛼𝑛^𝛼<12. (2.30) By Lemma 1.4, ^𝑛_𝑘=1𝑃𝑋_𝑘>𝜀𝑛^𝛼≤4𝑐𝑃max_{1≤𝑘≤𝑛}𝑋_𝑘>𝜀𝑛^𝛼,∀𝜀>0, (2.31) which together with (2.27), ^∞_𝑛=1𝑛^{𝑛𝛼𝑝−2}_𝑘=1𝑃𝑋_𝑘>𝜀𝑛^𝛼<∞,∀𝜀>0. (2.32) By 𝑋_𝑘≻𝑋, we obtain 𝐸|𝑋|^𝑝≪^∞_𝑛=1𝑛^𝛼𝑝−1𝑃|𝑋|>𝜀𝑛^𝛼<∞. (2.33) This completes the proof of Theorem 2.2.

Acknowledgments

The author is very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. Supported by the National Natural Science Foundation of China (11061012), and project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ([2011] 47), and the support program of Key Laboratory of Spatial Information and Geomatics (1103108-08).