Sufficient and necessary conditions of convergence for ρ͠ mixing random variables

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.


. Some notations and known results
Let {Xn , n ≥ } be a sequence of random variables and if there exist a positive constant C (C ) and a random variable X, such that the left-hand side (right-hand side) of the following inequalities is satis ed for all n ≥ and x ≥ , then the sequence {Xn , n ≥ } is said to be weakly lower (upper) bounded by X. The sequence {Xn , n ≥ } is said to be weakly bounded by X if it is both weakly lower and upper bounded by X. A sequence of random variables {Un , n ≥ } is said to converge completely to a constant C if The concept of complete convergence was introduced rstly by Hsu and Robbins [13]. In view of the Borel-Cantelli lemma, complete convergence implies that Un → C almost surely. The complete moment convergence is a more general concept than the complete convergence, which was introduced by Chow [14]. Let {Zn , n ≥ } be a sequence of random variables and an > , bn > , q > , if ∞ n= anE{b − n |Zn| − ϵ} q + < ∞, for some or all ϵ > , 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 < r < , r ≤ p. Suppose that {Xn , n ≥ } is a sequence of i.i.d. random variables with mean zero, then E|X | p < ∞ is equivalent to the condition that ∞ n= n p/r− P |Sn| > εn /r < ∞, for all ε > , (1.5) and also equivalent to the condition that ∞ n= n p/r− P max ≤k≤n |S k | > εn /r < ∞, for all ε > . (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 di erence, negatively associated, mixing random variables and so on. In the present paper, we are interested in the ρ-mixing random variables.
For identically distributed ρ-mixing random variables, Peligrad and Gut [8] extended the results of Baum and Katz to ρ-mixing random variables (see Theorem B); subsequently, An and Yuan [4] extended the results of Peligrad and Gut [8] to weighted sums of ρ-mixing random variables (see Theorem C); Gan [10] obtained a su cient condition on complete convergence (see Theorem D). Theorem B. (Peligrad and Gut [8]) Let {Xn , n ≥ } be a sequence of identically distributed ρ-mixing random variables, αp > , α > / , and suppose that EX = for α ≤ . Assume that limn→∞ ρn < , then E|X | p < ∞ is equivalent to the condition that ∞ n= n αp− P max ≤j≤n |S j | > εn α < ∞, for all ε > . (1.7) Theorem C. ( Theorem D. (Gan [10]) Let {Xn , n ≥ } be a sequence of identically distributed ρ-mixing random variables with ρ( ) < and < p ≤ , δ > , α ≥ max{( + δ)/p, }. If EX = and E|X | p < ∞, then ∞ n= n αp− −δ P |Sn| > εn α < ∞, for all ε > . (1.9) In this paper, the purpose is to study and establish the equivalent conditions on complete convergence and complete moment convergence for ρ-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 di erent in various places. Let I(A) be the indicator function of the set A, an = O(bn) represent an ≤ Cbn for all n ≥ .

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 ( , ∞), is said to be slowly varying at in nity if lim Firstly, we state the complete convergence for the weighted sums of {Xn , n ≥ }. Since log x ∈ L and ∈ L, we can obtain the following corollaries.

Proofs of Main results . Some lemmas
To prove our results, we rst give some lemmas as follows.
Lemma 3.1. (Kuzmaszewska [19]) Let {Xn , n ≥ } 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 > , then for any t > and n ≥ , the following statements hold: [5]) Let {Xn , n ≥ } be a sequence of ρ-mixing random variables, then there exists a positive constant C such that for any x ≥ and all n ≥ , [20]) Let {Yn , n ≥ } and {Zn , n ≥ } be sequences of random variables, then for any q > , ϵ > and all a > , we have

Lemma 3.3. (Sung
(3.5) [9]) For a positive integer N ≥ and positive real numbers q ≥ and ≤ r < , there is a positive constant C = C(q, N, r) such that if {Xn , n ≥ } is a sequence of random variables with ρ N ≤ r, with EX k = and E|X k | q < ∞ for every k ≥ , then for all n ≥ , (iv) C nr l(ϵ n ) ≤ n k= kr l(ϵ k ) ≤ C nr l(ϵ n ) for every r > , ϵ > and positive integer n.

. Proof of Theorem 2.1
Without loss of generality, we can assume that a ni > for all ≤ i ≤ n, n ≥ . For xed n, let We will consider the following three cases, p > , p = and < p < respectively.
(ii) Let p = . By the similar proof of (3.9), we have max ≤j≤n j i= a ni EX ni → .
(iii) Let < p < . Since X i = X ni + X ni for all i ≥ , we have By Lemma 3.1 and Lemma 3.5, we obtain that ≤CE|X| p l(|X| /α ) < ∞.