Convergence of series of normally distributed random variables

Question

$X_1, X_2,...$ are independent and each $X_n$ has normal distribution $N(0,a^n)$ for $a>0$. I need to find probability that the series $$\sum_{n=1}^\infty X_n$$ converges. How do I approach this problem?

Answer 1

Hints: the series converges with probability $0$ or $1$ by $0-1$ law.

If $a <1$ then $\sum E|X_n| <\infty$ so the series converges with probability $1$.

If $a=1$ the series converges with probability $0$. In fact no i.i.d. series converges except when the terms are all $0$ with probability $1$.

When $a>1$ it is easy to see that $|X_n|$ tends to $\infty$ in probability which implies that the series cannot converges with probability $1$ and hence it converges with probability $0$.

You can also solve this problem using Kolmogorov's 3-series theorem.