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

摘要

Let { X n k , u n ≤ k ≤ v n , n ≥ 1 } and { A n k , u n ≤ k ≤ v n , n ≥ 1 } be two arrays of random variables defined on the same probability space ( Ω , A , P ) and B n be sub-σ-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 n k } with exponent r and conditionally strongly residually h-integrable relative to B n concerning the array { A n k } 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.