The Annals of Probability

Toward a General Law of the Iterated Logarithm in Banach Space

Uwe Einmahl

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Abstract

A general bounded law of the iterated logarithm for Banach space valued random variables is established. Our results implies: (a) the bounded LIL of Ledoux and Talagrand, (b) a bounded LIL for random variables in the domain of attraction of a Gaussian law and (c) new LIL results for random variables outside the domain of attraction of a Gaussian law in cases where the classical norming sequence $\{\sqrt{nLLn}\}$ does not work. Basic ingredients of our proof are an infinite-dimensional Fuk-Nagaev type inequality and an infinite-dimensional version of Klass's $K$-function.

Article information

Source
Ann. Probab., Volume 21, Number 4 (1993), 2012-2045.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989009

Digital Object Identifier
doi:10.1214/aop/1176989009

Mathematical Reviews number (MathSciNet)
MR1245299

Zentralblatt MATH identifier
0790.60034

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
Bounded law of the iterated logarithm $K$-function LIL behavior randomization Rademacher random variables

Citation

Einmahl, Uwe. Toward a General Law of the Iterated Logarithm in Banach Space. Ann. Probab. 21 (1993), no. 4, 2012--2045. doi:10.1214/aop/1176989009. https://projecteuclid.org/euclid.aop/1176989009


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