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 ( , A, P) and B n be sub- -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.