The Annals of Probability
- Ann. Probab.
- Volume 8, Number 6 (1980), 1003-1036.
Almost Sure Invariance Principles for Partial Sums of Mixing $B$-Valued Random Variables
The approximation of partial sums of $\phi$-mixing random variables with values in a Banach space $B$ by a $B$-valued Brownian motion is obtained. This result yields the compact as well as the functional law of the iterated logarithm for these sums. As an application we strengthen a uniform law of the iterated logarithm for classes of functions recently obtained by Kaufman and Philipp (1978). As byproducts we obtain necessary and sufficient conditions for an almost sure invariance principle for independent identically distributed $B$-valued random variables and an almost sure invariance principle for sums of $d$-dimensional random vectors satisfying a strong mixing condition.
Ann. Probab., Volume 8, Number 6 (1980), 1003-1036.
First available in Project Euclid: 19 April 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60F15: Strong theorems
Secondary: 60G17: Sample path properties 60B05: Probability measures on topological spaces 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60B10: Convergence of probability measures
Kuelbs, J.; Philipp, Walter. Almost Sure Invariance Principles for Partial Sums of Mixing $B$-Valued Random Variables. Ann. Probab. 8 (1980), no. 6, 1003--1036. doi:10.1214/aop/1176994565. https://projecteuclid.org/euclid.aop/1176994565