Equivalent conditions of complete moment convergence for extended negatively dependent random variables

In this paper, we study the equivalent conditions of complete moment convergence for sequences of identically distributed extended negatively dependent random variables. As a result, we extend and generalize some results of complete moment convergence obtained by Chow (Bull. Inst. Math. Acad. Sin. 16:177-201, 1988) and Li and Spătaru (J. Theor. Probab. 18:933-947, 2005) from the i.i.d. case to extended negatively dependent sequences.


Introduction
Random variables X and Y are said to be negative quadrant dependent (NQD) if for all x, y ∈ R. A collection of random variables is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the collection satisfies (.).
It is important to note that (.) implies for all x, y ∈ R. Moreover, it follows that (.) implies (.), and hence (.) and (.) are equivalent. However, Ebrahimi and Ghosh ( []) showed that (.) and (.) are not equivalent for a collection of three or more random variables. Accordingly, the following definition is needed to define sequences of extended negatively dependent random variables.
Definition . Random variables X  , . . . , X n are said to be extended negatively dependent (END) if there exists a constant M >  such that for all real x  , . . . , x n , P n j= (X j ≤ x j ) ≤ M n j= P(X j ≤ x j ), P n j= (X j > x j ) ≤ M n j= P(X j > x j ).
An infinite sequence of random variables {X n ; n ≥ } is said to be END if every finite subset X  , . . . , X n is END.
Definition . Random variables X  , X  , . . . , X n , n ≥ , are said to be negatively associated (NA) if for every pair of disjoint subsets A  and A  of {, , . . . , n}, where It is easy to see from the definitions that NA implies ND and END. But example . in Wu and Jiang ( []) shows that ND or END does not imply NA. Thus, it is shown that END is much weaker than NA. In the articles listed earlier, a number of well-known multivariate distributions are shown to possess the END properties. In many statistical and mechanic models, an END assumption among the random variables in the models is more reasonable than an independent or NA assumption. Because of wide applications in multivariate statistical analysis and reliability theory, the notions of END random variables have attracted more and more attention recently. ). Hence, it is highly desirable and of considerable significance in the theory and application to study the limit properties of END random variables theorems and applications.
Chow ( []) first investigated the complete moment convergence, which is more exact than complete convergence. Thus, complete moment convergence is one of the most important problems in probability theory. Recent results can be found in Chen and Wang ). In addition, Li and Spătaru ( []) obtained the following complete moment convergence theorem: Let {X, X n ; n ≥ } be a sequence of independent and identically distributed (i.i.d.) random variables with partial sums S n = n i= X i , n ≥ . Suppose that r ≥ ,  < p < , q > . Then Furthermore, Chen and Wang ( []) showed that (.) and are equivalent.

Conclusions
The purpose of this paper is to study and establish the equivalent conditions of complete moment convergence of the maximum of the absolute value of the partial sum max ≤k≤n |S k | for sequences of identically distributed extended negatively dependent random variables. Our results not only extend and generalize some results on the complete moment convergence such as obtained by Chow ( []) and Li and Spătaru ( []) from the i.i.d. case to extended negatively dependent sequences, but also from partial sums case to the maximum of partial sums. Our research results and research methods provide some useful ideas and methods for the study of the complete moment convergence of the maximum of partial sums for other dependent random variables. 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 denote that there exists a constant c >  such that a n ≤ cb n for sufficiently large n, ln x means ln(max(x, e)) and I denotes an indicator function.
Theorem . Let {X, X n ; n ≥ } be a sequence of identically distributed END random variables with partial sums S n = n i= X i , n ≥ . Suppose that r > ,  < p < , q >  and EX =  for  ≤ p < . Then the following statements are equivalent:

Proofs
The following three lemmas play important roles in the proof of our theorems. Lemma . can be obtained directly from the definition of END sequences.
Further, if P(max ≤k≤n |X k | > x) →  as n → ∞, then there exists a positive constant c such that for all n ≥ , Let x ≥ n /p , r - < α <  and an integer N > max((r -)(αr -) - , q(αrp) - ). Define, for  ≤ k ≤ n, n ≥ , It is obvious that S k =  j= S (j) k . Hence, in order to establish (.), it suffices to prove that By combining this with (.), Markov's inequality and we get From the definition of X () k , it is clear that X () k > . Thus, by Definition . and (.), Hence, by the definition of N , -αrpN + q - < - and -(αr -)N + r - < -, Similarly, we can show In order to estimate J () , we first verify that When  < p < , by Markov's inequality and E|X| p < ∞, When  ≤ p < , by EX =  and E|X| rp < ∞, we get That is, (.) holds. Hence, in order to prove J () < ∞, it suffices to prove that Obviously, X () k is increasing on X k , thus by Lemma ., {X () k ; k ≥ } is also a sequence of END random variables. In view of Lemma ., taking u = max(rp, q) and If u < , then by E|X| u < ∞ and |X ()  | ≤ x α , That is, (.) holds.
Finally, we prove that (.) ⇒ (.). By (.) and max ≤k≤n |X k | ≤  max ≤k≤n |S k |, it follows that it implies that P(max  j ≤k< j+ |X k | >  j/p x) → , j → ∞ for any x > . Thus, by Lemma ., for any x > , there is c >  such that for sufficiently large j  j P |X| >  j/p x ≤ cP max  j ≤k< j+ |X k | >  j/p x .