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The law of the iterated logarithm in uniformly convex Banach spaces


Author: Michel Ledoux
Journal: Trans. Amer. Math. Soc. 294 (1986), 351-365
MSC: Primary 60B12; Secondary 60B11
DOI: https://doi.org/10.1090/S0002-9947-1986-0819953-X
MathSciNet review: 819953
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Abstract: We give necessary and sufficient conditions for a random variable $ X$ with values in a uniformly convex Banach space $ B$ to satisfy the law of the iterated logarithm. Precisely, we show that a $ B$-valued random variable $ X$ satisfies the (compact) law of the iterated logarithm if and only if $ E\{ \vert\vert X\vert{\vert^2}/{L_2}\vert\vert X\vert\vert\} < \infty $, the family $ \{ \vert{x^{\ast}}(X){\vert^2};\,{x^{\ast}} \in {B^{\ast}},\,\vert\vert{x^{\ast}}\vert\vert = 1\} $ is uniformly integrable and $ {S_n}/{a_n} \to 0$ in probability.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0819953-X
Keywords: Law of the iterated logarithm, uniformly convex Banach spaces, smooth norm
Article copyright: © Copyright 1986 American Mathematical Society

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