2008
DOI: 10.1080/07362990802007186
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On Complete Convergence for Arrays of Random Elements

Abstract: A complete convergence theorem for arrays of rowwise independent random variables was obtained by Kruglov, Volodin, and Hu (Statistics and Probability Letters 2006Letters , 76:1631Letters -1640. In this article, we extend the result to a Banach space without any additional conditions. No assumptions are made concerning the geometry of the underlying Banach space.

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Cited by 5 publications
(4 citation statements)
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“…In this note, we show that Theorem A holds without Downloaded by [University of Bristol] at 04:28 03 November 2014 condition (1.4), and also establish the rate of convergence for maxima of partial sums without centering. The proofs are quite different from Sung et al [10] and Hu et al [5].…”
Section: Introductionmentioning
confidence: 91%
See 1 more Smart Citation
“…In this note, we show that Theorem A holds without Downloaded by [University of Bristol] at 04:28 03 November 2014 condition (1.4), and also establish the rate of convergence for maxima of partial sums without centering. The proofs are quite different from Sung et al [10] and Hu et al [5].…”
Section: Introductionmentioning
confidence: 91%
“…One of the differences is that the conclusions of Theorems B and C deal with maxima of partial sums. But the main difference appears in Sung et al [10], where the authors required the centering by expectations of truncated random variables. In this note, we show that Theorem A holds without Downloaded by [University of Bristol] at 04:28 03 November 2014 condition (1.4), and also establish the rate of convergence for maxima of partial sums without centering.…”
Section: Introductionmentioning
confidence: 98%
“…Our proof of Theorem 3.1 is different from that of Theorem 1 of [8] (which relied on the Rosenthal [17] inequality) and Theorem 3.1 of [15] (which relied on Theorem 3 of [13]). …”
Section: Introductionmentioning
confidence: 92%
“…This result has been generalized and extended in several directions by a number of authors; for results up to 1999, see the discussions in Hu, Rosalsky, Szynal, and Volodin [3]. More recent work on complete convergence is that of Hu and Volodin [4], Hu, Li, Rosalsky, and Volodin [5], Hu, Ordóñez Cabrera, Sung, and Volodin [6], Kuczmaszewska [7], Sung, Volodin, and Hu [8], Kruglov, Volodin, and Hu [9], Sung and Volodin [10], Hernández, Urmeneta, and Volodin [11], Sung, Ordóñez Cabrera, and Hu [12], Sung, Urmeneta, and Volodin [13], Chen, Hernández, Urmeneta, and Volodin [14], and Hu, Rosalsky, and Wang [15]. Some of these generalizations and extensions concern a Banach space setting: a sequence of Banach space valued random elements is said to converge completely to the 0 element of the Banach space if the corresponding sequence of norms converges completely to 0.…”
Section: Introductionmentioning
confidence: 98%