An application of the Ryll-Nardzewski-Woyczynski 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[greater-or-equal, slanted]1} (which are not necessarily independent) with the series [summation operator]n=1[infinity] Vn converging unconditionally in probability and an infinite array a={ani, i[greater-or-equal, slanted]n, n[greater-or-equal, slanted]1} of constants, conditions are given under which (i) for all n[greater-or-equal, slanted]1, the sequence of weighted sums [summation operator]i=nm aniVi converges in probability to a random element Tn(a) as m-->[infinity], and (ii) 6 uniformly in a as n-->[infinity] 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 Woyczynski (1975, Proc. Amer. Math. Soc. 53, 96-98).