I was working through J.R. Norris Markov Chains and got stuck at exercise 2.3.1. The question is the following:
Let $S_1, S_2, \ldots$ be independent exponential random variables of parameters $\lambda_1, \lambda_2, \ldots$ respectively. Show that $\lambda_1 S_1$ is exponential with parameter $1$.
Use the strong law of large numbers to show, first in the special case $\lambda_n = 1$ for all $n$, and then subject only to the condition $\sup_n \lambda_n < \infty$, that $$ \mathbb{P}\left(\sum_{n=1}^{\infty} S_n = \infty\right) = 1 $$ Is the condition $\sup_n \lambda_n < \infty$.
I was able to prove that $\lambda_i S_i \sim \exp(1)$, but I was not able to prove that $\mathbb{P}\left(\sum_{n=1}^{\infty} S_n = \infty\right) = 1 $, even for the special case $\lambda_n = 1 \, \forall n$.
Intuitively I have concluded the following: Since $\mathbb{E}(S_n) = \frac{1}{\lambda_n}$ and $\sup_n \lambda_n < \infty$, we can conclude that $$ \mathbb{E}(\sum_{n=1}^{\infty} S_n) = \sum_{n=1}^{\infty} \mathbb{E}(S_n) = \sum_{n=1}^{\infty} \frac{1}{\lambda_n} = \infty, $$ since $\sup_n \lambda_n < \infty$.
Is there any way to build on this intuition? How do I exactly use the strong law of large numbers to solve the exercise?
Thanks in advance!
First let $\lambda_n = 1$. Then, by the strong law of large numbers, $$\frac{1}{N} \sum_{n=1}^{N} S_n \to 1$$ almost surely. In particular, the sequence of partial sums $(\sum_{n=1}^{N} S_n)_{N\geq 1}$ is almost surely unbounded (otherwise the limit would be $0$ on a set of nonzero probability).
Now assume $\Lambda:=\sup_n \lambda_n < \infty$. By what we've just shown and because $\lambda_i S_i \sim \exp(1)$, we know that $$\mathbb{P}\left(\sum_{n=1}^{\infty}\lambda_n S_n = \infty\right) = 1.$$
But if $\sum_{n=1}^{\infty}\lambda_n S_n (\omega) = \infty$, then also $\frac{1}{\Lambda}\sum_{n=1}^{\infty}\lambda_n S_n (\omega) = \infty$, and
$$\infty = \sum_{n=1}^{\infty}\frac{\lambda_n}{\Lambda} S_n \leq \sum_{n=1}^{\infty} S_n .$$