Double stochastic integrals, random quadratic forms and random series in Orlicz spaces (Q1091038)

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 \(\iint f(s,t)dX(s)dX(t)\) 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 \(\ell_{\psi}\) valued sequences \((a_ i)\) for which the series \(\sum a_ i\xi_ i\) converges a.s. in \(\ell_{\psi}\). The above three problems are closely related.