The Annals of Probability

A Strong Convergence Theorem for Banach Space Valued Random Variables

J. Kuelbs

Full-text: Open access

Abstract

We prove a strong convergence theorem for Banach space valued random variables. One corollary of this result establishes necessary and sufficient conditions for the law of the iterated logarithm (LIL) in the Banach space setting. We also prove an exact generalization of the Hartman-Wintner law of the iterated logarithm provided the random variables involved take values in a real separable Hilbert space or some other Banach space with smooth norm.

Article information

Source
Ann. Probab., Volume 4, Number 5 (1976), 744-771.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995982

Digital Object Identifier
doi:10.1214/aop/1176995982

Mathematical Reviews number (MathSciNet)
MR420771

Zentralblatt MATH identifier
0365.60034

JSTOR
links.jstor.org

Subjects
Primary: 60B05: Probability measures on topological spaces
Secondary: 60B10: Convergence of probability measures 60F10: Large deviations 28A40

Keywords
Measurable norm Gaussian measure law of the iterated logarithm differentiable norm submartingale Berry-Esseen estimates

Citation

Kuelbs, J. A Strong Convergence Theorem for Banach Space Valued Random Variables. Ann. Probab. 4 (1976), no. 5, 744--771. doi:10.1214/aop/1176995982. https://projecteuclid.org/euclid.aop/1176995982


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