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Teoriya Veroyatnostei i ee Primeneniya, 1981, Volume 26, Issue 3, Pages 584–590
(Mi tvp2603)
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This article is cited in 15 scientific papers (total in 15 papers)
The laws of large numbers for identically distributed Banach space valued random variables
T. A. Azlarov, N. A. Volodin Taškent
Abstract:
Let $0<\alpha<2$ and let $B_{\alpha}$ be an arbitrary Banach space if $0<\alpha\le 1$ and $B_{\alpha}$ be an $\alpha$-type space if $1<\alpha<2$ (definition of $\alpha$-type space see [1]); let $B_{\alpha}$ be separable when $\alpha\ge 1$. Without loss of generality we suppose that $\mathbf EX=0$ if $\mathbf E\|X\|<\infty$ where $X$ is Banach space valued random variable.
Theorem.{\it Let $0<\alpha<2$ and $\{X_n\}$ be a sequence of independent identically distributed $B_{\alpha}$-valued random variables, $S_n=X_1+\dots+X_n$. The following conditions are equivalent.}
I. $\mathbf E\|X_1\|^\alpha<\infty$.
II. $\|n^{-1/\alpha}S_n\|\to 0$ a. s., $n\to\infty$.
III. $\mathbf E\|S_n\|^{\alpha}=o(n)$, $n\to\infty$.
IV. $\displaystyle\sum_{n=1}^{\infty} n^{-1}\mathbf P\{\|S_n\|>\varepsilon n^{1/\alpha}\}<\infty$
for every $\varepsilon>0$.
Received: 06.02.1979
Citation:
T. A. Azlarov, N. A. Volodin, “The laws of large numbers for identically distributed Banach space valued random variables”, Teor. Veroyatnost. i Primenen., 26:3 (1981), 584–590; Theory Probab. Appl., 26:3 (1982), 573–580
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https://www.mathnet.ru/eng/tvp2603 https://www.mathnet.ru/eng/tvp/v26/i3/p584
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