## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 3 (1985), 885-897.

### Convergence of Quadratic Forms in $p$-Stable Random Variables and $\theta_p$-Radonifying Operators

Stamatis Cambanis, Jan Rosinski, and Wojbor A. Woyczynski

#### Abstract

Necessary and sufficient conditions are given for the almost sure convergence of the quadratic form $\sum \sum f_{jk}M_jM_k$ where $(M_j)$ is a sequence of i.i.d. $p$-stable random variables. A connection is established between the convergence of the quadratic form and a radonifying property of the infinite matrix operator $(f_{kj})$.

#### Article information

**Source**

Ann. Probab., Volume 13, Number 3 (1985), 885-897.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992912

**Digital Object Identifier**

doi:10.1214/aop/1176992912

**Mathematical Reviews number (MathSciNet)**

MR799426

**Zentralblatt MATH identifier**

0575.60018

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E07: Infinitely divisible distributions; stable distributions

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

**Keywords**

Stable random variables quadratic forms $\theta_p$-radonifying operators

#### Citation

Cambanis, Stamatis; Rosinski, Jan; Woyczynski, Wojbor A. Convergence of Quadratic Forms in $p$-Stable Random Variables and $\theta_p$-Radonifying Operators. Ann. Probab. 13 (1985), no. 3, 885--897. doi:10.1214/aop/1176992912. https://projecteuclid.org/euclid.aop/1176992912