Let and be two arrays of random variables defined on the same probability space and be sub-σ-algebras of . Let be a constant. In this paper, we introduce some concepts of conditional residual h-integrability such as conditionally residually h-integrable relative to concerning the array with exponent r and conditionally strongly residually h-integrable relative to concerning the array 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.
MSC: 60F15, 60F25.