I am trying to solve the following excercise.
Let $\{X_i\}$ be a sequence of independent stable random variables, that is $X_i \sim S_\alpha (\sigma_i,\beta_i,\mu_i)$ with $0<\alpha \leq 2$. The series $\sum_{i=1}^\infty X_i$ converges a.s. if and only if $\sum_{i=1}^\infty \mu_i$ converges and $\sum_{i=1}^\infty \sigma_i^{\alpha} < \infty$.
I tried to solve with the Kolmogorov's three series theorem, but I am lost. I will appreciate any suggestions and helps. Thanks!