A Gaussian version of Littlewood's theorem on random power series

Abstract

We prove a Littlewood-type theorem on random analytic functions for not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space $H^2(\dd)$ by a Gaussian process whose covariance matrix $K$ induces a bounded operator on $l^2$, then the resulting random function is almost surely in $H^p(\dd)$ for any $p>0$. The case $K=\text{Id}$, the identity operator, recovers Littlewood's theorem. A new ingredient in our proof is to recast the membership problem as the boundedness of an operator. This reformulation enables us to use tools in functional analysis and is applicable to other situations. The sharpness of the new condition and several ramifications are discussed.