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July, 1988 Characterization of the Law of the Iterated Logarithm in Banach Spaces
M. Ledoux, M. Talagrand
Ann. Probab. 16(3): 1242-1264 (July, 1988). DOI: 10.1214/aop/1176991688


Using a Gaussian randomization technique, we prove that a random variable $X$ with values in a Banach space $B$ satisfies the (compact) law of the iterated logarithm if and only if (i) $E(\|X\|^2/LL\|X\|) < \infty$, (ii) $\{|\langle x^\ast, X \rangle |^2; x^\ast \in B^\ast, \|x^\ast\| \leq 1\}$ is uniformly integrable and (iii) $S_n(x)/a_n\rightarrow 0$ in probability. In particular, if $B$ is of type 2, in order that $X$ satisfy the law of the iterated logarithm it is necessary and sufficient that $X$ have mean zero and satisfy (i) and (ii). The proof uses tools of the theory of Gaussian random vectors as well as by now classical arguments of probability in Banach spaces. It also sheds some light on the usual law of the iterated logarithm on the line.


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M. Ledoux. M. Talagrand. "Characterization of the Law of the Iterated Logarithm in Banach Spaces." Ann. Probab. 16 (3) 1242 - 1264, July, 1988.


Published: July, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0662.60008
MathSciNet: MR942766
Digital Object Identifier: 10.1214/aop/1176991688

Primary: 60B12
Secondary: 46B20 , 60B11 , 60G15

Keywords: ‎Banach spaces , Gaussian randomization type 2 , Law of the iterated logarithm

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 3 • July, 1988
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