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

For a sequence of Banach space valued random elements {Vn,n≥1} (which are not necessarily independent) with the series ∑n=1 ∞Vn converging unconditionally in probability and an infinite array a={ani,i≥n,n≥1} of constants, conditions are given under which (i) for all n≥1, the sequence of weighted sums ∑i=n maniVi converges in probability to a random element Tn(a) as m→∞, and (ii) Tn(a)→P0 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). © 2000 Elsevier Science B.V