Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

Abstract

For a sequence of constants {an, n ≥ 1}, an array of rowwise independent and stochastically dominated random elements {Vnj, j ≥ 1, n ≥ 1} in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space, and a sequence of positive integer-valued random variables {Tn, n ≥ 1}, a general weak law of large numbers of the form ∑Tn j = 1 aj(Vnj-cnj)/b[αn] →p 0 is established where {cnj, j ≥ 1, n ≥ 1}, αn → ∞, bn → ∞ are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {Vnj, j ≥ 1, n ≥ 1}. Illustrative examples include one wherein the strong law of large numbers fails.