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

Let 1 <= p < 2, alpha > p, {a(ni), 1 <= i <= n, n >= 1} be a set of real numbers with the property sup(n >= 1) , n(-1) Sigma(n)(i=1) vertical bar a(ni)vertical bar(alpha) < infinity and let {X, X-n, n >= 1} be a sequence of H-valued rho*-mixing random vectors coordinatewise stochastically upper dominated by a random vector X. We provide conditions such that for any epsilon > 0 the following inequalities hold: Sigma(infinity)(n=1)n(-1)P(max(1 <= k <= n)parallel to Sigma(k)(i=1)a(ni)X(i)parallel to > epsilon n(1/p)) < infinity, Sigma(infinity)(n=1)n(-1-1/p) E (max(1 <= k <= n) parallel to Sigma(k)(i=1)a(ni)X(i)parallel to - epsilon n(1/p))(+) < infinity. These results generalize the results of Chen and Sung (cf. J. Ineq. Appl. 121, 1-16 (2018)) to the rho*-mixing random vectors in H. In addition, a Marcinkiewicz-Zygmund type strong law of rho*-mixing random vectors in H is presented.