## The Annals of Probability

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

### A Strong Convergence Theorem for Banach Space Valued Random Variables

#### 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