Random power series near the endpoint of the convergence interval

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.