대한수학회논문집 (Communications of the Korean Mathematical Society)
- 제10권2호
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- Pages.439-442
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- 1995
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- 1225-1763(pISSN)
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- 2234-3024(eISSN)
A NOTE ON SUMS OF RANDOM VECTORS WITH VALUES IN A BANACH SPACE
- Hong, Dug-Hun (Department of Statistics, Taegu Hyosung Catholic University) ;
- Kwon, Joong-Sung (Department of Mathematics, Sun Moon University)
- 발행 : 1995.04.01
초록
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].