• The Journal of the Korea Contents Association
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• v.6 no.5
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• pp.29-34
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• 2006

For the almost certainly convergent series $S_n=\sum_{i=1}^nV-i$ of independent random elements in Banach spaces, by investigating tail series laws of large numbers, the rate of convergence of the series $S_n$ to a random variable s is studied in this paper. More specifically, by studying the duality between the limiting behavior of the tail series $T_n=S-S_{n-1}=\sum_{i=n}^{\infty}V-i$ of random variables and that of Banach space valued random elements, an alternative way of proving a result of the previous work, which establishes the equivalence between the tail series weak law of large numbers and a limit law, is provided in a Banach space setting.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.439-442
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• 1995

Let ${X_n : n = 1,2,\cdots}$ be a sequence of pairwise independent identically distributed random vectors taking values in a separable Hilbert space H such that $E \Vert X_1 \Vert = \infty$. Let $S_n = X_1 + X_2 + \cdots + X_n$ and for any real $\alpha$ with $0 < \alpha < 1$ define a sequence ${\gamma_n(\alpha)}$ as $\gamma_n(\alpha) = inf {r : P(\Vert S_n \Vert \leq r) \geq \alpha}$. Then $$ lim_{n \to \infty} sup \Vert S_n \Vert/\gamma_n(\alpha) = \infty $$ holds. This is a generalization of Vvedenskaya[2].