15.7.1 Limit theorems

Problem: Let \( \xi_{1}, \xi_{2}, \ldots \) be a sequence of independent random variables, each of which has the Poisson distribution. Prove that the series \( \sum_{n=1}^{\infty} \xi_{n} \) almost surely converges only when the series \( \sum_{n=1}^{\infty} E \xi_{n} \) converges.