The law of the iterated logarithm in uniformly convex Banach spaces
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- by Michel Ledoux PDF
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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 $E\{ ||X|{|^2}/{L_2}||X||\} < \infty$, the family $\{ |{x^{\ast }}(X){|^2}; {x^{\ast }} \in {B^{\ast }}, ||{x^{\ast }}|| = 1\}$ is uniformly integrable and ${S_n}/{a_n} \to 0$ in probability.References
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Additional Information
- © Copyright 1986 American Mathematical Society
- 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