A probabilistic approach to strong natural boundaries

Abstract

We study the local non-extendability of random power series beyond their disk of convergence. We show that random power series formed by independent coefficients which are asymptotically anti-concentrated admit the circle of radius of convergence as strong natural boundary, even in a Nevanlinna sense. Our results extend previous work of Breuer and Simon (2011) for the case of independent coefficients. Our motivation stems from the study of Pad\'e approximants of random power series as a denoising tool.