Limiting Behavior of the Maximum of the Partial Sum for Linearly Negative Quadrant Dependent Random Variables under Residual Cesàro Alpha-Integrability Assumption

Linearly negative quadrant dependence is a special dependence structure. By relating such conditions to residual Cesaro alpha-integrability assumption, as well as to strongly residual Cesaro alpha-integrability assumption, some 𝐿𝑝-convergence and complete convergence results of the maximum of the partial sum are derived, respectively.


Introduction
The classical notion of uniform integrability of a sequence {X n } n∈N of integrable random variables is defined through the condition lim a → ∞ sup n≥1 E|X n |I |X n | > a 0. Landers and Rogge 1 proved that the uniform integrability condition is sufficient in order that a sequence of pairwise independent random variables verifies the weak law of large numbers WLLNs .Chandra 2 weakened the assumption of uniform integrability to Cesáro uniform integrability CUI and obtained L 1 -convergence for pairwise independent random variables.
Chandra and Goswami 3 improved the above-mentioned result of Landers and Rogge 1 .They showed that for a sequence of pairwise independent random variables, CUI is sufficient for the WLLN to hold and strong Cesáro uniform integrability SCUI is sufficient for the strong law of large numbers SLLNs to hold.Landers and Rogge 4 obtained a slight improvement over the results of Chandra 2 and Chandra and Goswami 3 for the case of nonnegative random variables.They showed that, in this case, the condition of pairwise independence can be replaced by the weaker assumption of pairwise nonpositive correlation.
Chandra and Goswami 5 introduced a new set of conditions called Cesáro α-integrability CI α and strong Cesáro α-integrability SCI α for a sequence of random

Preliminaries
First let us specify the two special kinds of uniform integrability we are dealing with in the subsequent sections, which were introduced by Chandra and Goswami 6 .
We point out that, {|X n | p } n∈N is SRCI α for any α > 0, provided that {X n } n∈N is stochastically dominated by a nonnegative random variable X with EX p δ < ∞ for some p ≥ 1 and δ > 0.
The condition of SRCI α is a "strong" version of the condition of RCI α .Moreover, for any α > 0, RCI α is strictly weaker than CI α , thereby weaker than CUI, while SRCI α is strictly weaker than SCI α , thereby much weaker than SCUI.
Next, we turn our attention to the dependence structure for random variables.For our purpose, we have to mention a special kind of dependence, namely, negative quadrant dependence.
Definition 2.3 cf.Lehmann 8 .Two random variables X and Y are said to be negative quadrant dependent NQD, in short if for any x, y ∈ R, P X < x, Y < y ≤ P X < x P Y < y .

2.3
A sequence {X n } n∈N of random variables is said to be pairwise NQD if X i and X j are NQD for all i, j ∈ N and i / j.The concept of LNQD sequence was introduced by Newman 9 .Some applications for LNQD sequence have been found; see, for example, the work by Newman 9 who established the central limit theorem for a strictly stationary LNQD process.Wang and Zhang 10 provided uniform rates of convergence in the central limit theorem for LNQD sequence.Ko et al. 11 obtained the Hoeffding-type inequality for LNQD sequence.Ko et al. 12 studied the strong convergence for weighted sums of LNQD arrays.Fu and Wu 13 studied the almost sure central limit theorem for LNQD sequences, and so forth.We note that " " means "O." Lemma 2.6 cf.Lehmann 8 .Let random variables X and Y be NQD.Then This lemma is easily proved by the results of Zhang 15 and Yuan and Wu 7 .Here we omit the details of the proof.Lemma 2.9.Let {X k } k∈N d be a centered LNQD random field.Then for any p > 1, there exists a positive constant c such that 7 This lemma is due to Zhang 15, Lemma 3.3 .Finally, we give a lemma which supplies us with the analytical part in the proofs of theorems in the subsequent sections.
for every n ≥ 1.

Residual Ces áro Alpha-Integrability and L p -Convergence of the Maximum of the Partial Sum
Let p > 1, and let h x be a strictly positive function defined on 1, ∞ .In this section, we discuss L p -convergence of the form of n  For our purpose, it suffices to prove 3.4 Using Lemma 2.8, the Hölder inequality, relation 3.2 , and the second condition in 2.1 of the RCI α property of the sequence {|X n | p } n∈N , we obtain

3.6
Using the first condition of 2.1 of the RCI α property of the sequence {|X n | p } n∈N , the last expression above clearly goes to 0 as n → ∞, from 1 < p < 2 and α < 1/ 2 − p , thus completing the proof.
Remark 3.2.Let 1 < p < 2, and let {X n } n∈N be a LNQD sequence of random variables.If Compared with Theorem 3.1, this result, whose proof can be completed by using Lemma 2.9, drops the maximum of the partial sum at the price of enlarging 1/n into 1/n

3.8
Proof of Theorem 3.3.By Lemma 2.7 and the H ölder inequality, 3.9 The proof is completed.

Strongly Residual Ces áro Alpha-Integrability and Complete Convergence of the Maximum of the Partial Sum
A sequence of random variables {X n } n∈N is said to converge completely to a constant a if for any ε > 0, In this case we write X n → a completely.This notion was given by Hsu and Robbins 16 .
Note that the complete convergence implies the almost sure convergence in view of the Borel-Cantelli lemma.The condition of SRCI α is a strong version of the condition of RCI α .In this section, we will show that each of the theorems in the previous section has a corresponding "strong" analogue in the sense of complete convergence.

4.2
Proof of Theorem 4.1.For any n ≥ 1, let m m n be the integer such that 2 m−1 < n ≤ 2 m .Observe that

4.3
Hence it suffices to show that Let Y n , Z n , S 1 n , and S 2 n be defined as in the proof of Theorem 3.1.We first prove that Using Lemma 2.8, the Hölder inequality, relation 3.2 , and the second condition in 2.1 of the RCI α property of the sequence By Lemma 2.7 and the H ölder inequality,

4.8
In view of the first condition in 2.1 of the RCI α property of the sequence 4.9 The last series above converges since α ∈ 0, 1/ 2 − p implies −1 2 − p α < 0, and therefore 4.7 holds.This completes the proof.
For the case p ≥ 2, we have the following result.

Lemma 2 .
10 cf.Landers and Rogge 4 .For sequences {a n } n∈N and {b n } n∈N of nonnegative real numbers, if

Theorem 4 . 2 . 1 n n i 1 10 then for any δ > 1/ 2 n 11 Proof of Theorem 4 . 2 . 0 E 2 1 i
Let p ≥ 2, and let {X n } n∈N be a LNQD sequence of random variables.If {X n } n∈N satisfies sup n≥1 E|X i | p < ∞, 4.−δ max 1≤i≤n |S i − ES i | −→ 0 completely.4.Let m n , n ≥ 1 be defined as in the proof of Theorem 4.1.Proceeding in the proof of 4.3 , we see that it suffices to show that Indeed by Lemma 2.7 and the H ölder inequality, ∞ m −mδ max 1≤i≤2 m |S i − ES i | −pδ−1 p/2 E|X i | p .
Definition 2.4 cf.Newman 9.A sequence{X n } n∈N of random variables is said to be linearly negative quadrant dependent LNQD, in short if for any disjoint subsets A, B ∈ Z and positive r j s, k∈A r k X k , j∈B r j X j are NQD.2.4 Remark 2.5.It is easily seen that if {X n } n∈N is a sequence of LNQD random variables, then {aX n b} n∈N is still a sequence of LNQD random variables, where a and b are real numbers.
Let {X n } n∈N be LNQD random variables sequences with mean zero.Then for 1 < p < 2, there exists a positive constant c such that and g are both nondecreasing (or both nonincreasing) functions, then f X and g Y are NQD.Lemma 2.7 cf. Hu et al. 14 .Let {X n } n∈N be a LNQD sequence of random variables with EX n 0. Assume that there exists a p > 2 satisfying E|X i | p < ∞ for every i ≥ 1.Then, there exists a positive constant c such that p,p/2 is a positive constant depending only on p.It is easily seen that when p 2, the above equation still holds true.Lemma 2.8.
−h p max 1≤i≤n |S i − ES i | for a LNQD sequence {X n } n∈N of random variables, provided that {|X n | p } n∈N is RCI α for an appropriate condition.Our first result is dealing with the case 1 < p < 2. Let 1 < p < 2, and let {X n } n∈N be a LNQD sequence of random variables.If {|X n | p } n∈N is RCI α for some α ∈ 0, 1/ 2 − p , then n −1 max 1≤i≤n n ≥ 1, and define, for each n ≥ 1,Z n X n − Y n , S Z i .It is easy to see that |Y n | min{|X n |, n α }, |Z n | |X n | − n α I |X n | > n α, and|Z n | p ≤ |X n | p − n α I |X n | p >n α 3.2 for all p > 1.Note that, for each n ≥ 1, Y n and Z n are monotone transformations of the initial variable X n .This implies that LNQD assumption is preserved by this construction in view of Lemma 2.6.Precisely, {Y n − EY n } n∈N and {Z n − EZ n } n∈N are also LNQD sequences of zero mean random variables.
1/p .Next we consider the case p ≥ 2. Let p ≥ 2, and let {X n } n∈N be a LNQD sequence of random variables.If {X n } n∈N satisfies