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

Tien-Chung Hu, Eunwoo Nam, Andrew Rosalsky and Andrei I. Volodin

Statistics & Probability Letters, 2000, vol. 48, issue 4, 369-374

Abstract: 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).

Keywords: Real; separable; Banach; space; Weighted; sums; of; random; elements; Converge; unconditionally; in; probability; Converge; in; probability; Tail; series; Uniform; weak; law; of; large; numbers (search for similar items in EconPapers)
Date: 2000
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