Conditional convergence for randomly weighted sums of random variables based on conditional residual h-integrability

Let {X-nk, u(n) <= k <= v(n), n >= 1} and {A(nk), u(n) <= k <= v(n), n >= 1} be two arrays of random variables defined on the same probability space (Omega, A, P) and B-n be sub-sigma-algebras of A. Let r > 0 be a constant. In this paper, we introduce some concepts of conditional residual h-integrability such as conditionally residually h-integrable relative to B-n concerning the array {A(nk)} with exponent r and conditionally strongly residually h-integrable relative to B-n concerning the array {A(nk)} with exponent r. These concepts are more general than some known setting of randomly weighted sums of random variables. Based on the conditions of conditional residual h-integrability with exponent r and conditional strongly residual h-integrability with exponent r, we obtain the conditional mean convergence and conditional almost sure convergence for randomly weighted sums.