Double Stochastic Integrals, Random Quadratic Forms and Random Series in Orlicz Spaces

Abstract

Let $X(t), t \geq 0$, be a process with independent, symmetric and stationary increments and let $(\xi_i)$ be i.i.d. symmetric real random variables. We provide a characterization of functions $f(s, t), s, t \geq 0$, such that the double integral $\int\int f(s, t) dX(s) dX(t)$ exists, a characterization of infinite matrices $(\alpha_{ij})$ such that the double series $\sum\alpha_{ij}\xi_i\xi_j$ converges a.s. and a characterization of Orlicz space $l_\psi$ valued sequences $(a_i)$ for which the series $\sum a_i\xi_i$ converges a.s. in $l_\psi$. The above three problems are closely related.