Radius of convergence of a series of random variables

Question

Let $X_n$ be i.i.d. and (a.s.) bounded random variables.(none of them identically zero) Prove that the radius of convergence of the series with coefficients $X_n$, $f(\omega,t)=\sum_{n=0}^{+\infty}X_n(\omega)t^n$ is exactly $1$. (Hint:use Borell-Cantelly lemma.)

Answer 1

Let $0\le t<1$, then $\forall \omega \in \Omega$ $$ \sum_{i=0}^n |X_i(\omega)| t^n \le M \sum_{i=0}^n t^n \le \frac{M}{1-t} $$ Thus the series is absolutely convergent (increasing and bounded) with a radius of convergence $R\ge 1$.