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
✍ Scribed by Tien-Chung Hu; Eunwoo Nam; Andrew Rosalsky; Andrei I. Volodin
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 95 KB
- Volume
- 48
- Category
- Article
- ISSN
- 0167-7152
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✦ Synopsis
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 inÿnite array a = {ani; i¿n; n¿1} of constants, conditions are given under which (i) for all n¿1, the sequence of weighted sums m i = n aniVi converges in probability to a random element Tn(a) as m → ∞, and (ii) Tn(a) P → 0 uniformly in a as n → ∞ where a is in a suitably restricted class of inÿnite arrays. The key tool used in the proof is a theorem of Ryll-Nardzewski and Woyczyà nski (1975, Proc. Amer. Math. Soc. 53, 96 -98).