## The Annals of Probability

### Weak and $L^p$-Invariance Principles for Sums of $B$-Valued Random Variables

Walter Philipp

#### Abstract

Suppose that the properly normalized partial sums of a sequence of independent identically distributed random variables with values in a separable Banach space converge in distribution to a stable law of index $\alpha$. Then without changing its distribution, one can redefine the sequence on a new probability space such that these partial sums converge in probability and consequently even in $L^p (p < \alpha)$ to the corresponding stable process. This provides a new method to prove functional central limit theorems and related results. A similar theorem holds for stationary $\phi$-mixing sequences of random variables.

#### Article information

Source
Ann. Probab., Volume 8, Number 1 (1980), 68-82.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994825

Digital Object Identifier
doi:10.1214/aop/1176994825

Mathematical Reviews number (MathSciNet)
MR556415

Zentralblatt MATH identifier
0426.60033

JSTOR
Philipp, Walter. Weak and $L^p$-Invariance Principles for Sums of $B$-Valued Random Variables. Ann. Probab. 8 (1980), no. 1, 68--82. doi:10.1214/aop/1176994825. https://projecteuclid.org/euclid.aop/1176994825
• See Correction: Walter Philipp. Correction: Correction to "Weak and $L^p$-Invariance Principles for Sums of $B$-Valued Random Variables". Ann. Probab., Volume 14, Number 3 (1986), 1095--1101.