Let $X_n$ be a sequence of random variables such that $X_i \sim Poi(\lambda_i)$. If $X_n$ converges to $0$ almost surely show that $\lim_n \lambda_n = 0$.
If $X_n$ converges to $0$ almost surely, this implies
- $P(\lim_n X_N = 0) = 1$
Since the expectation of a poisson random variable is just its parameter, I was thinking of saying the parameters $\lambda_n$ must run to $0$ by the SLLN but the strong law only applies to a sample mean so now I have no idea where to start.