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

TL;DR

This paper extends Littlewood's theorem to Gaussian processes with bounded covariance operators, showing that randomized functions in Hardy spaces almost surely belong to all $H^p$ spaces, using operator theory techniques.

Contribution

It introduces a Gaussian version of Littlewood's theorem for dependent processes and recasts the problem as an operator boundedness issue, broadening the theorem's applicability.

Findings

Randomized functions are almost surely in all $H^p$ spaces for $p>0$

The boundedness of the covariance operator $K$ is key to membership results

The approach generalizes classical results using functional analysis tools

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.