Almost sure convergences of series of Poisson random variables Let $X_i$ be independent Poisson with mean $\lambda_i$. Show that 
i)$\sum_1^\infty\lambda_i<\infty$ implies $\sum_1^\infty X_i$ converges almost surely to a finite limit.
ii)$\sum_1^\infty\lambda_i=\infty$ implies $\sum_1^\infty X_i=\infty$ almost surely
I could prove (i) by using Kolmogorov one series lemma just setting $Y_i=X_i-\lambda_i$, but I could not prove the second part. Any help/hint/solution for (ii) is appreciated. Thanks in advance
 A: *

*$$\infty > \sum_{n} \lambda_n = \sum_{n} E[X_n] = E[\sum_{n} X_n]$$
$$\to \infty > \sum_{n} X_n$$


*$$E[X_n] = Var[X_n] = \lambda_n$$


Use contrapositive of Kolmogorov's three-series theorem to show that $\forall A > 0$, one of the three series diverges.
A: A calculation of Laplace transformation of Poisson $\big(\lambda_i\big)$ shows us , $\forall u \in\mathbb{R}_+$,
$$\mathbb{E}\Big[ \,  e^{-u\cdot X_i} \, \Big] = \exp\big[\,  -\lambda_i\cdot\big(1- e^{-u}\big)  \,\big] \, \,\Longrightarrow\quad \mathbb{E}\Big[\,  s^{X_i} \, \Big] =  \exp\big[\,  -\lambda_i\cdot\big(1- s\big)  \,\big]\, \text{with $s = e^{-u}\in\big(0,1\big)$},$$
then it follows from dominated convergence and finite-additivity of Poisson distributions that
   \begin{align*}
   \mathbb{E}\left\{ \,\,  s^{\sum_{i\geq 1}X_i } \,\,\right\}  &=   \mathbb{E}\left\{ \,\,  \lim_{k\to+\infty}s^{\sum_{i = 1}^k X_i } \,\,\right\}  =  \lim_{k\to+\infty}  \mathbb{E}\left\{ \,\, s^{\sum_{i = 1}^k X_i } \,\,\right\}  =  \lim_{k\to+\infty}\exp\left(\,\, -\sum_{i=1}^k \lambda_i\cdot(1-s)\,\,  \right) \\
    &=  \exp\left(\,\, -(1-s)\sum_{i\geq1} \lambda_i\,\,  \right)\,\,,\quad\text{for $s\in(0,1)$.}
  \end{align*}
So  the sum of an independent sequence of Poisson random variables is  Poisson  with parameter equal to the sum of their rates. And if the sum of rates$=\infty$,  then $\sum_{i=1}^\infty X_i = +\infty$ a.s.
