Random power series near the endpoint of the convergence interval

Abstract

In this paper, we are going to consider power series $$\sum _{n=1}^{\infty }{a}_n{x}^n,$$ 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 $$\underset{x\to 1-}{\lim }\sum _{n=1}^{\infty }{a}_n{x}^n=\infty (\text{resp. }\underset{x\to 1-}{\lim }\sum _{n=1}^{\infty }{a}_n{x}^n=-\infty )$$ with probability $1$. Also, if the expected value of the coefficients is $0$, then $$\underset{x\to 1-}{\mathrm{limsup}}\sum _{n=1}^{\infty }{a}_n{x}^n=\infty ,\underset{x\to 1-}{\mathrm{liminf}}\sum _{n=1}^{\infty }{a}_n{x}^n=-\infty$$ with probability $1$. We investigate the analogous question in terms of Baire categories.