A probabilistic approach to strong natural boundaries

TL;DR

The paper investigates when random power series with independent coefficients fail to extend beyond their circle of convergence, focusing on strong natural boundaries on the unit circle. It develops a weighted anti-concentration framework, leveraging Rogozin’s inequality and small-ball estimates to show that if the coefficients exhibit asymptotic anti-concentration (in a precise weighted sense), then the unit circle is almost surely a strong natural boundary in the Nevanlinna sense. The results encompass symmetric coefficients, extend Breuer–Simon’s bounded-coefficient regime via Berry–Esseen, and yield consequences for arcwise log-integrability and local boundary behavior. These findings have implications for the spectral analysis of Padé approximants of random power series and broaden the classical understanding of natural boundaries in random analytic functions. The work thus provides a probabilistic mechanism for SNB, linking anti-concentration properties to rigidity of boundary behavior with potential applications in denoising and random series analysis.

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 Padé approximants of random power series as a denoising tool.

Theorems & Definitions (38)

• Theorem 1.1: Ryll-Nardzewski, 1953
• Theorem 1.2: Breuer, Simon, 2011
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 2.1
• Definition 2.2: Lévy concentration function
• Lemma 2.3: Rogozin, 1961