probability - Limit of Poisson Distribution

Just for fun, I'm looking at the concentration of the Poisson Distribution near it's mean. For $\lambda=10$, there is a 36% probability of being within 10% of the mean. For $\lambda=100$, that number jumps to 70%, and at $\lambda = 1000$, it's nearly 100%.

I've just taken a course in Probability...and I'm trying to make it more concrete in my mind. I'm just looking for a way to formalize this observation, so I can relate it to what I've learned.

It seems I have for $X_n$ Poisson w/ Parameter $n$, $P\big(\big|\frac{X_n}{n}-1\big| > \epsilon\big) \rightarrow 0$ as $n \rightarrow \infty$.

Is that the strongest thing I can say? How would you say this in the language of probability? It's almost like convergence in probability, except for the division by $n$. Thanks!

Answer

This is a case of the Weak Law of Large Numbers, since $X_n$ is the sum of $n$ independent Poisson(1) random variables. A stronger statement is given by the
Central Limit Theorem. See also the theory of Large Deviations, which tells you
that for $a > 1$,

$$\lim_{n \to \infty} \dfrac{1}{n} \log P(X_n/n \ge a ) = - a \ln(a) + a - 1$$

(if I've done my calculations right).