Complete qth moment convergence for arrays of random variables

Let {X-ni, 1 <= i <= n, n >= 1} be an array of random variables with EXni = 0 and E vertical bar X-ni vertical bar(q) < infinity for some q >= 1. For any sequences {a(n), n >= 1} and {b(n), n >= 1} of positive real numbers, sets of sufficient conditions are given for complete qth moment convergence of the form Sigma(infinity)(n=1) n = 1 b(n)a(n)(a) E(max(1 <= k <= n) Sigma(k)(i=1) X-ni vertical bar - epsilon a(n))(+)(q) < infinity,. for all is an element of> 0, where x(+) = max{x, 0}. From these results, we can easily obtain some known results on complete qth moment convergence.