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

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 $(f_{kj})$.