An application of the Ryll-Nardzewski–Woyczyński theorem to a uniform weak law for tail series of weighted sums of random elements in Banach spaces

Abstract

For a sequence of Banach space valued random elements { V n , n⩾1} (which are not necessarily independent) with the series ∑ n=1 ∞ V n converging unconditionally in probability and an infinite array a={a ni, i⩾n, n⩾1} of constants, conditions are given under which (i) for all n⩾1, the sequence of weighted sums ∑ i=n m a niV i converges in probability to a random element T n ( a) as m→∞, and (ii) T n(a) → P 0 uniformly in a as n→∞ where a is in a suitably restricted class of infinite arrays. The key tool used in the proof is a theorem of Ryll-Nardzewski and Woyczyński (1975, Proc. Amer. Math. Soc. 53, 96–98).