## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60B10: Convergence of probability measures

**Keywords**

Invariance principles domains of attraction stable laws Banach space valued random variables mixing sequences of random variables

#### Citation

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

#### Corrections

- 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.Project Euclid: euclid.aop/1176992465