I need the following result in my bachelor's thesis, and want to better understand the proof of it.
$X_{n}$ independent and $X_n \sim \mathcal{P}(n) $ meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to \infty} \frac{X_{n}}{n}$ almost surely ?
This is from this question: Find the almost sure limit of $X_n/n$, where each random variable $X_n$ has a Poisson distribution with parameter $n$
The answers were helpful, still there are two things I don't understand:
(the more important one) Math1000, in his/her answer uses the Chernoff bound, but I don't know how they use it/which version, to get to the estimated result. If I use the definition of the bound from wikipedia, I get a different result.
How do we get from the lim inf to the limit, i.e. why do we know that the limit exists?
Help would be much appreciated
PS: would have commented on that thread, but I don't have enough reputation for that yet.