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? 



Convergence 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 $X\_n = X + Y\_n$.


