We have encountered the following problem that we think that should be true. Let $\{X_n\}_{n\geq 0}$ a sequence of random variables which we know that $\mathbb{E}[X_n]$ tends to infinity.
The question is the following: can we assure that the sequence does NOT converge in distribution to a Poisson?
No. It's easy to construct a sequence $Y_n$ with $Y_n \to 0$ a.s. but $E Y_n \to +\infty$. (You can even have $E Y_n \equiv +\infty$ if you wish.) Now let $X$ be a fixed Poisson random variable and $X_n = X + Y_n$.
As Nate Eldridge noted, the answer is no. For a positive result, you need some extra condition. Suppose e.g. the variances $\text{Var}(X_n) < c (E X_n)^2$ for $n$ sufficiently large, with some constant $c \in (0,1)$. Then $X_n$ can't converge in distribution.
