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Let {Xnk, un ? k ? vn, n ? 1} and {Ank, un ? k ? vn, n ? 1} be two arrays of random variables defined on the same probability space (-,A, P) and Bn 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 Bn concerning the array {Ank} with exponent r and conditionally strongly residually h-integrable relative to Bn concerning the array {Ank} 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. © 2013 Shen et al.

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