Let $p(x) = \sum_{k \geq 0} a_k x^k$ where the $a_k$'s are IID random variables taken from a mean-zero random variable taking finitely many values in $\mathbb{R}$; it clearly converges for $-1<x<1$. Is it a.s. true that the sign of $p(x)$ oscillates infinitely often as $x \rightarrow 1^-$? That is, is it the case (with probability 1) that there exist $x_1 < x_2 < x_3 < \dots$ in (0,1) such that $p(x_k)$ has the same sign as $(-1)^k$?

I imagine that this is well-known for $P(1) = P(-1) = 1/2$ (where $P(r)$ denotes the probability that $a_k = r$); the case that most interests me is $P(1) = P(-1) = 1/4$ and $P(0) = 1/2$, but I'm sure that the same technique settles both cases.

I am also interested in knowing about the magnitude of the oscillations.