Mean Convergence Theorems with or without Random Indices for Randomly Weighted Sums of Random Elements in Rademacher Type p Banach Spaces

摘要

Some mean convergence theorems are established for randomly weighted sums of the form ∑(A{sub}(nj)V{sub}(nj)) (j from 1 to k{sub}n) and ∑(A{sub}(nj)V{sub}(nj)) (j from 1 to T{sub}n) where {A{sub}(nj), j≥1, n≥1} is an array of random variables, {V{sub}(nj), j≥1, n≥1} is an array of mean 0 random elements in a separable real Rademacher type p (1≤p≤2) Banach space, and {k{sub}n, n≥1} and {T{sub}n, n≥1} are sequences of positive integers and positive integer-valued random, variables, respectively.