How to determine $\lim_{n \to \infty}P(X_1+\cdot\cdot\cdot+X_n \leq n)$?

Question

Let $(X_n)_n$ be a sequence of iid Poisson-distributed random variables with parameter $\lambda >0$. How can I calculate $$\lim_{n \to \infty}P(X_1+\cdot\cdot\cdot+X_n \leq n)?$$

Some help would be much appreciated.

Answer 1

This is $$P\left(\frac{X_1+\cdots+X_n}n\le1\right).$$ If $\lambda=1$ the limit is $\frac12$ by the Central Limit Theorem. Otherwise $\frac1n(X_1+\cdots+X_n)$ has mean $\lambda$ and variance $\lambda/n$. This variance tends to zero, so the limit is $1$ for $\lambda<1$ and $0$ for $\lambda>1$. You can use Chebyshev's inequality to prove this.