## The Annals of Probability

- Ann. Probab.
- Volume 7, Number 4 (1979), 738-743.

### The Almost Sure Stability of Quadratic Forms

James M. Wilmesmeier and F. T. Wright

#### Abstract

Let $w_{jk}$ be a doubly indexed sequence of weights, let $\{X_k\}$ be a sequence of independent random variables and let $Q_n = \Sigma^n_{j,k=1} w_{jk}X_jX_k$. Sufficient conditions for the almost sure stability of $Q_n$ are given and the "tightness" of these conditions is investigated. These quadratic forms are weighted sums of dependent variables; however, their stability properties are very much like those established in the literature for weighted sums of independent variables.

#### Article information

**Source**

Ann. Probab., Volume 7, Number 4 (1979), 738-743.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994995

**Digital Object Identifier**

doi:10.1214/aop/1176994995

**Mathematical Reviews number (MathSciNet)**

MR537219

**Zentralblatt MATH identifier**

0411.60033

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G50: Sums of independent random variables; random walks

**Keywords**

Stability quadratic forms degenerate convergence almost sure convergence

#### Citation

Wilmesmeier, James M.; Wright, F. T. The Almost Sure Stability of Quadratic Forms. Ann. Probab. 7 (1979), no. 4, 738--743. doi:10.1214/aop/1176994995. https://projecteuclid.org/euclid.aop/1176994995