Abstract
In this paper, We study the complete convergence and Lp- convergence for the maximum of the partial sum of negatively superadditive dependent random vectors in Hilbert space. The results extend the corresponding ones of Ko (Ko, 2020) to H-valued negatively superadditive dependent random vectors.
Highlights
Alam and Saxena [1] introduced the concept of negative association as follows: a finite family of random variables Xi, 1 ≤ i ≤ n is said to be negatively associated (NA) for every pair of disjoint subsets of A and B of {1, 2, . . . , n}Cov(f(Xi, i ∈ A)g(Xj, j ∈ B)) ≤ 0, whenever f and g are coordinate-wise nondecreasing functions and the covariance exists
Based on the above superadditive function, the concept of negatively superadditive dependent (NSD) random variables was introduced by Hu [3] as follows: a random vector X (X1, X2, . . . , Xn) is said to be negatively superadditive dependent (NSD) if Eφ(X1, X2, . . . , Xn) ≤ Eφ(X1∗, X2∗, . . . , Xn∗ ), where X1∗, X2∗, . . . , Xn∗ are independent such that Xi∗ and Xi have the same distribution for each i and φ is a superadditive function such that the above expectations exist
Hu [3] gave an example for illustrating that negatively superadditive dependence (NSD) does not imply negative association (NA), and Christofides and Vaggelatou [4] indicated that NA implies NSD
Summary
Alam and Saxena [1] introduced the concept of negative association as follows: a finite family of random variables Xi, 1 ≤ i ≤ n is said to be negatively associated (NA) for every pair of disjoint subsets of A and B of {1, 2, . . . , n}Cov(f(Xi, i ∈ A)g(Xj, j ∈ B)) ≤ 0, whenever f and g are coordinate-wise nondecreasing functions and the covariance exists. Based on the above superadditive function, the concept of negatively superadditive dependent (NSD) random variables was introduced by Hu [3] as follows: a random vector X As the concept of H-valued NA random vectors was introduced by Ko et al [5], Son et al [9] presented the concept of H-valued negatively superadditive dependent (NSD) random vectors as follows: a sequence Xn, n ≥ 1 of H-valued random vectors is said to be NSD if for any d ≥ 1, the sequence (X(n1), X(n2), . X(nd)), n ≥ 1 of Rd-valued random vectors is negatively superadditive dependent (NSD). We extend Lemma 2 to a sequence of Hilbert valued random vectors as follows. (i) If E‖X‖r < ∞, (1/n) ni 1 E‖Xi‖r ≤ CE‖X‖r (ii) (1/n) ni 1 E‖X′i‖r ≤ C(E‖X′‖r) + ArP(‖X‖ > A)
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