The complete moment convergence for CNA random vectors in Hilbert spaces

In this paper we establish the complete moment convergence for sequences of coordinatewise negatively associated random vectors in Hilbert spaces. The result extends the complete moment convergence in (Ko in J. Inequal. Appl. 2016:131, 2016) to Hilbert spaces as well as generalizes the Baum-Katz type theorem in (Huan et al. in Acta Math. Hung. 144(1):132-149, 2014) to the complete moment convergence.

Ko et al. [] proved almost sure convergence for H-valued NA random vectors and Thanh [] proved almost sure convergence for H-valued NA random vectors and provided extensions of the results in Ko et al. []. Miao [] showed Hajeck-Renyi inequality for NA random vectors in a Hilbert space.
Huan et al. [] presented another concept of negative association for H-valued random vectors which is more general than the concept of H-valued NA random vectors introduced by Ko et al. [] as follows.
A sequence {X n , n ≥ } of H-valued random vectors is said to be coordinatewise negatively associated (CNA) if, for each j ≥ , the sequence {X Let {X n , n ≥ } be a sequence of random variables. Let {a n , n ≥ } and {b n , n ≥ } be sequences of positive numbers and q > . The concept of complete moment convergence is introduced as follows. If ∞ n= a n E{b - n |X n | -} Let {X, X n , n ≥ } be a sequence of H-valued random vectors. We consider the following inequalities: where X (j) n = X n , e j and X (j) = X, e j for all j ≥ . If there exists a positive constant C  (C  ) such that the left-hand side (right-hand side) of (.) is satisfied for all j ≥ , n ≥  and t ≥ , then the sequence {X n , n ≥ } is said to be coordinatewise weakly lower (upper) bounded by X. The sequence {X n , n ≥ } is said to be coordinatewise weakly bounded by X if it is both coordinatewise weakly lower and upper bounded by X.
In this paper we show the complete moment convergence for CNA random vectors in Hilbert spaces. The result extends the complete moment convergence for NA random variables in R  (the main result in Ko []) to a Hilbert space as well as generalizes the Baum-Katz type theorem (Theorem . in Huan et al. []) for CNA random vectors in a Hilbert space to the complete moment convergence in a Hilbert space.

Preliminaries
The key tool for proving our results is the following maximal inequality.
Lemma . (Kuczmaszewska []) Let {X n , n ≥ } be a sequence of random variables weakly upper bounded by a random variable X. Let r >  and, for some A > , and Then, for some constant C > , According to the proof of Theorem . in Huan et al.
For I  , by the Markov inequality, (.), (.) and the fact that E|Y | p = p ∞  y p- P(|Y | > y) dy, we obtain For I  , we estimate that ni , i ≥ } is NA for all j ≥ , and so {Y ni , i ≥ } is CNA. Hence, by the Markov inequality, Lemma . and Lemma .(ii), we have The last inequality above is obtained by Lemma .(ii).
For I  , by (.) we have that I  < ∞. For I  , by a standard calculation we observe that It remains to prove I  < ∞. From (.), (.), (.) and the fact that EX (j) i = , for all i ≥  and j ≥ , we obtain By (.) we have that I  < ∞. For I  , by a standard calculation as in (.), we obtain which yields I  < ∞, together with I  < ∞. Hence, the proof is completed.
The following lemma shows that Lemmas . and . still hold under a sequence of identically distributed H-valued CNA random vectors with zero means. The following section will show that the complete moment convergence for NA random variables in Ko [] can be extended to a Hilbert space.

Main results
The proofs of main results can be obtained by using the methods of the proofs as in the main results of Ko [].
Theorem . Let r and α be positive numbers such that  ≤ r <  and αr > . Let {X n , n ≥ } be a sequence of H-valued CNA random vectors with zero means. If {X n , n ≥ } is coordinatewise weakly upper bounded by a random vector X satisfying (.), then we obtain where a + = max{a, }.
Proof The proof can be obtained by a similar calculation in the proof of Theorem . of Ko []. From Lemmas . and . we obtain  The following theorem shows that complete convergence and complete moment convergence still hold under a sequence of identically distributed H-valued CNA random vectors with zero means.

Conclusions
. In Section  we have obtained the complete moment convergence for a sequence of mean zero H-valued CNA random vectors which is coordinatewise weakly upper bonded by a random variable and the related results. . Theorem . generalizes the complete convergence for a sequence of mean zero H-valued CNA random vectors in Huan et al.
[] to the complete moment convergence.