Deterministic radius of convergence of power series

Question

Let $X_i$'s be i.i.d. random variables. Consider the random power series

\begin{equation*} \sum_{n=0}^\infty = X_n z^n \end{equation*}

Is there any deterministic (almost surely) radius of convergence of the above series in the following two cases (a): $P(X_i = 1) = P(X_i = -1)$ = 1/2 (b): $X_i$ follows N(0; 1). If so, find the radius.

Answer 1

For each $\omega\in\Omega$ we have $R(\omega)=1/\limsup_{n\to\infty}|X_n(\omega)|^{1/n}= 1$, in case (a) (obviously) and also (only slightly less obviously, according to Robert) in case (b), with probability 1.