A logarithmic criterion for the convergence of multiparameter random series (Q1074945)

The stated aim of this paper is to give a unified approach to the convergence of random power series. The author considers linear isomorphisms \(T_ 1,...,T_ d\) of a separable Banach space X such that \(\lim_{n\to \infty}\| T^ n_ j\| =0,\) \(j=1,2,...,d.\) For \(n=(n_ 1,...,n_ d)\in {\mathbb{N}}^ d\), define \(T_ n=T_ 1^{n_ 1}T_ 2^{n_ 2}...T_ d^{n_ d}\), and let \(Z_ n\) be an X-valued random vector. Assume that the \(Z_ n\) are mutually independent and share the distribution of a fixed random vector Z. The author's main theorem then asserts that the random series \(\sum_{n}T_ n(Z_ n)\) is a.s. convergent if and only if \(E(\log^ d(1+\| Z\|))<\infty\).