A Gaussian version of Littlewood’s theorem for random power series

Abstract

We prove a Littlewood-type theorem for random analytic functions associated with not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space H 2 ( D ) H^2(\mathbb {D}) by a Gaussian process whose covariance matrix K K induces a bounded operator on l 2 l^2 , then the resulting random function is almost surely in H p ( D ) H^p(\mathbb {D}) for any p > 0 p>0 . The case K = I d K=\mathrm {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.