On the Weak Law with Random Indices for Arrays of Banach Space Valued Random Elements

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 Banach space for some p ∈ [1, 2], 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)/bαn P− → 0 is established, where {cnj, j ≥ 1, n ≥ 1} is an array of truncated expectations, and αn → ∞, bn → ∞ are suitable sequences. No assumption is made con-cerning the existence of expected values or absolute moments of the random elements {Vnj, j ≥ 1, n ≥ 1}. The current work is a new version of a result of Adler, Rosalsky, and Volodin (J. Theoret. Probab. vol. 10, 1997, 605–623)