## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 2 (1984), 390-426.

### Convergence of Sums of Mixing Triangular Arrays of Random Vectors with Stationary Rows

#### Abstract

This paper deals with the convergence in distribution to Gaussian, generalized Poisson and infinitely divisible laws of the row sums of certain $\phi$ or $\psi$-mixing triangular arrays of Banach space valued random vectors with stationary rows. Necessary and sufficient conditions for convergence in terms of individual r.v.'s are proved. These include sufficient conditions for the convergence to a stable law of the normalized sums of certain $\phi$-mixing, stationary sequences. An invariance principle for stationary, $\phi$-mixing triangular arrays is given.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 2 (1984), 390-426.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993297

**Digital Object Identifier**

doi:10.1214/aop/1176993297

**Mathematical Reviews number (MathSciNet)**

MR735845

**Zentralblatt MATH identifier**

0542.60012

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F17: Functional limit theorems; invariance principles

**Keywords**

Mixing triangular array Banach space valued random vector weak convergence of measures Gaussian measure $\tau$-centered Poisson measure infinitely divisible measure invariance principle

#### Citation

Samur, Jorge D. Convergence of Sums of Mixing Triangular Arrays of Random Vectors with Stationary Rows. Ann. Probab. 12 (1984), no. 2, 390--426. doi:10.1214/aop/1176993297. https://projecteuclid.org/euclid.aop/1176993297