On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables

and Applied Analysis 3 2. Preliminaries In this section, wewill present some important lemmaswhich will be used to prove the main results of the paper. The first three lemmas come from Sung [1]. Lemma 11 (cf.[1]). Let {a n , n ≥ 1} be a sequence of positive constants with a n /n ↑. Then the following properties hold. (i) {a n , n ≥ 1} is a strictly increasing sequence with an a n ↑ ∞. (ii) ∑∞ n=1 P(X > a n ) < ∞ if and only if∑∞ n=1 P(X > 2a n ) < ∞. (iii) ∑∞ n=1 P(X > a n ) < ∞ if and only if∑∞ n=1 P(X > αa n ) < ∞ for any α > 0. Lemma 12 (cf. [1]). If {a n , n ≥ 1} is a sequence of positive constants with a n /n ↑ and X is a random variable, then n a n E |X| I (|X| ≤ a n ) ≤ ∞ ∑ n=0 P (|X| > a n ) . (6) Lemma 13 (cf. [1]). Let {a n , n ≥ 1} be a sequence of positive constants with a n /n ↑ ∞ and X is a random variable. If ∑∞ n=1 P(|X| > a n ) < ∞, then (n/a n )E|X|I(|X| ≤ a n ) → 0. The next one is the basic property for pairwise NQD random variables, which was given by Lehmann [8] as follows. Lemma 14 (cf. [8]). Let X and Y be NQD; then (i) EXY ≤ EXEY; (ii) P(X > x, Y > y) ≤ P(X > x)P(Y > y), for any x, y ∈ R; (iii) if f and g are both nondecreasing (or nonincreasing) functions, then f(X) and g(Y) are NQD. The following one is the generalized Borel-Cantelli lemma, which was obtained by Matula [10]. Lemma 15 (cf. [10]). Let {A n , n ≥ 1} be a sequence of events. (i) If ∑∞ n=1 P(A n ) < ∞, then P(A n , i.o.) = 0. (ii) If P(A k A m ) ≤ P(A k )P(A m ) for k ̸ =m and ∑∞ n=1 P(A n ) = ∞, then P(A n , i.o.) = 1. With the generalized Borel-Cantelli lemma accounted for, we can establish the second Borel-Cantelli lemma for pairwise NQD random variables as follows. Corollary 16 (second Borel-Cantelli lemma for pairwise NQD random variables). Let {a n , n ≥ 1} be a sequence of positive constants with a n /n ↑. Let {X n , n ≥ 1} be a sequence of pairwise NQD random variables. Then X n a n 󳨀→ 0 a.s. ⇐⇒ ∞ ∑ n=1 P (󵄨󵄨󵄨Xn 󵄨󵄨󵄨 > an) < ∞. (7) Proof. “⇐”. By Lemma 11, ∑∞ n=1 P(|X n | > a n ) < ∞ is equivalent to ∑∞ n=1 P(|X n | > a n ε) < ∞ for all ε > 0, which yields thatX n /a n → 0 a.s. by Borel-Cantelli lemma. ⇒. Let X n /a n → 0 a.s., which implies that X+ n /a n → 0 a.s. andX n /a n → 0 a.s. For any ε > 0, denote

We point out that the keys to the proofs of the main results of Sung [1] are the Khintchine-Kolmogorov-type convergence theorem and the second Borel-Cantelli lemma for pairwise independent events (e.g., see Theorem 4.2.5 in [6] or Theorem 2.18.5 in [7]), while these are not proved for pairwise negatively quadrant dependent random variables (pairwise NQD, in short; see Definition 1).If we want to generalize the main results of Sung [1] to the case of pairwise NQD random variables, we should propose new methods or prove the Khintchine-Kolmogorov-type convergence theorem and the second Borel-Cantelli lemma for pairwise NQD random variables.The answer is positive.
Firstly, let us recall the concept of pairwise negatively quadrant dependent random variables as follows.
The concept of pairwise NQD random variables was introduced by Lehmann [8], which includes pairwise independent random sequence and some negatively dependent sequences, such as negatively associated sequences (see [9][10][11][12][13]), negatively orthant dependent sequences (see [9,[14][15][16][17][18]), and linearly negative quadrant dependent sequences (see [19][20][21]).Hence, studying the probability limiting behavior of pairwise NQD random variables and its applications in probability theory and mathematical statistics are of great interest.Many authors have dedicated themselves to the study of it.Matula [10] gained the Kolmogorov-type strong law of large numbers for the identically distributed pairwise NQD sequences; Wu [22] gave the generalized threeseries theorem for pairwise NQD sequences and proved the Marcinkiewicz strong law of large numbers; Chen [23] discussed Kolmogorov-Chung strong law of large numbers for the nonidentically distributed pairwise NQD sequences under very mild conditions; Wan [24] and Huang et al. [25] obtained the complete convergence for pairwise NQD random sequences; Wang et al. [26], Li and Yang [27], Gan and Chen [28], Shi [29], Xu and Tang [30], and Tang [31] studied the strong convergence properties for pairwise NQD random variables; Sung [21] established the   convergence for weighted sums of arrays of rowwise pairwise NQD random variables under weaker uniformly integrable conditions; and so on.The main purpose of the paper is to establish the second Borel-Cantelli lemma for pairwise NQD random variables and generalize the main results of Sung [1] to the case of pairwise NQD random variables without adding any extra conditions.
Our main results are as follows.The first two results are the complete convergence for pairwise NQD random variables.
The following two theorems are the results on strong convergence for pairwise NQD random variables.Theorem 4. Let {  ,  ≥ 1} be a sequence of positive constants with   / ↑.Let {,   ,  ≥ 1} be a sequence of pairwise NQD random variables with identical distribution.Then, the following statements are equivalent: Remark 7. Theorems 2 and 3 deal with the complete convergence for pairwise NQD random variables.Theorems 4 and 5 deal with the strong laws of large numbers for pairwise NQD random variables, which are equivalent to the mild condition ∑ ∞ =1 (|| >   ) < ∞.Pairwise NQD is a very wide dependence structure, which includes independent sequence as a special case.Hence, Theorems 2-5 generalize the corresponding ones for pairwise i.i.d.random variables to the case of pairwise NQD random variables.
Remark 8.Under the conditions of Theorem 3 and  2 ≤   , we can get the Marcinkiewicz-Zygmund-type strong law of large numbers for pairwise NQD random variables as follows: Hence, Etemadi's strong law of large numbers follows from Theorem 4 with   = .
Throughout the paper, let () be the indicator function of the set .  denotes a positive constant not depending on , which may be different in various places.Denote  0 = 0,  + =  ( ≥ 0), and  − = − ( < 0).

Preliminaries
In this section, we will present some important lemmas which will be used to prove the main results of the paper.
The first three lemmas come from Sung [1].
Lemma 12 (cf.[1]).If {  ,  ≥ 1} is a sequence of positive constants with   / ↑ and  is a random variable, then Lemma 13 (cf.[1]).Let {  ,  ≥ 1} be a sequence of positive constants with The next one is the basic property for pairwise NQD random variables, which was given by Lehmann [8] as follows.
The following one is the generalized Borel-Cantelli lemma, which was obtained by Matula [10].
With the generalized Borel-Cantelli lemma accounted for, we can establish the second Borel-Cantelli lemma for pairwise NQD random variables as follows.
The last one is the Kolmogorov-type strong law of large numbers for pairwise NQD random variables obtained by Chen [23], which plays an important role in proving the main results of the paper.
In the following, we will prove  2 < ∞.It is easily checked that Similar to the proof of  3 < ∞ in Theorem 2, we can get that  23 < ∞.
For fixed  ≥ 1, denote Similar to the proof of  2 < ∞ in Theorem 2, we have Since Var(  ) ≤  2  < ∞ for each  ≥ 1, we have by ( 23) and (24) and Lemma 17 that Note that which together with Kronecker's lemma yield that By ( 26) and ( 28), we have Therefore, the desired result (ii) follows by ( 29) and ( 30) immediately.
Next, we will prove that (ii) ⇒ (i).Assume that Therefore, (i) ⇔ (iii) follows by the statements above immediately.This completes the proof of the theorem.
Proof of Corollary 6.The techniques used here are the second Borel-Cantelli lemma for pairwise NQD random variables (see Corollary 16) and Theorem 5.The proof is similar to that of Corollary 2.1 of Sung [1], so the details of the proof are omitted.