Complete and Complete Moment Convergence of the Weighted Sums of $ρ^∗$-Mixing Random Vectors in Hilbert Spaces

Abstract

. Let 1 ≤ p < 2, α > p , { a ni ,1 ≤ i ≤ n , n ≥ 1 } be a set of real numbers with the property sup n ≥ 1 n − 1 P n i = 1 | a ni | α < ∞ and let { X , X n , n ≥ 1 } be a sequence of H -valued ρ ∗ -mixing random vectors coordinatewise stochastically upper dominated by a random vector X . We provide conditions such that for any ε > 0 the following inequalities hold: These results generalize the results of Chen and Sung (cf. J. Ineq. Appl. 121 , 1–16 (2018)) to the ρ ∗ -mixing random vectors in H . In addition, a Marcinkiewicz-Zygmund type strong law of ρ ∗ -mixing random vectors in H is presented. random vectors, complete moment convergence, weighted sums, Marcinkiewicz-Zygmund type strong law of large numbers.