Random power series near the endpoint of the convergence interval

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 $0$, 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.