Random power series near the endpoint of the convergence interval

TL;DR

This paper studies the behavior of random power series near the convergence endpoint, showing almost sure divergence to infinity or minus infinity depending on the expected coefficient value, and explores Baire category aspects.

Contribution

It establishes almost sure divergence results for random power series at the boundary, depending on the expected coefficient, and investigates their properties in the Baire category sense.

Findings

Series diverge to infinity if expected coefficient is positive.

Series diverge to minus infinity if expected coefficient is negative.

Series oscillate unboundedly if expected coefficient is zero.

Abstract

In this paper, we are going to consider power series $$ \sum_{n=1}^{\infty} a_nx^n, $$ where the coefficients $a_n$ are chosen independently at random from a finite set with uniform distribution. We prove that if the expected value of the coefficients is positive (resp. negative), then $$ \lim_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=\infty\qquad (\text{resp. }\lim_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=-\infty) $$ with probability $1$. Also, if the expected value of the coefficients is $0$, then $$ \limsup_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=\infty,\qquad \liminf_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=-\infty $$ with probability $1$. We investigate the analogous question in terms of Baire categories.