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