# Showing that a sequence of random variables with increasing expected value converges to a Poisson random variable

Consider a sequence $\{X_n\}_{n \geq 1}$ of nonnegative, integer-valued random variables. For any random variable $Y$ and $k \geq 1$, let $(Y)_k = Y(Y-1)(Y-2)\dots(Y-k+1)$ be the $k^\mathrm{th}$ factorial moment of $Y$. To begin with, suppose that there exists $\lambda \in \mathbb{R}_{> 0}$ such that the expected value $\mathbb{E}((X_n)_k)$ satisfies $$\mathbb{E}((X_n)_k) = \lambda^k(1+o(1))$$ for every $k$ (in the limit as $n \to \infty$). Then it is a well-known result (see Theorem 1.22 in Random Graphs by Bela Bollobas) that the random variables $X_n$ weakly converge to a Poisson random variable; i.e. we have that $$\lim_{n \to \infty} \mathbb{P}(X_n = \ell) = e^{-\lambda} \cdot \frac{\lambda^{\ell}}{\ell !},$$ where $\mathbb{P}(X_n = \ell)$ denotes the probability that $X_n = \ell$.

Now, suppose that for every $k$ we have $$\mathbb{E}((X_n)_k) = n^k(1 + o(1)),$$ where by $o(1)$ I mean something that decays at least as fast as $1/n$. (Notice here that for each $k$, the expected value $\mathbb{E}((X_n)_k)$ grows with $n$, whereas previously this expected value was bounded independent of $n$.) In this new situation, my guess is that for large $n$, the random variable $X_n$ might be approximately Poisson with parameter $\lambda = n$, but I am not sure whether this is a well-known result or how to approach proving this type of claim rigorously. Any advice to this end is appreciated!

• What do you think the random variables look like? Can you find any example of such random variables, and see what they converge to? If you think it converges to a Poisson limit, what's the parameter $\lambda$ of the limit? You should think about these things before asking this type of question. – Will Sawin Jul 26 '15 at 22:44
• I edited my post to describe what I think the random variables look like in the limit of large $n$; from the examples I am working with, I believe they have parameter $\lambda = n$, but I am not sure how one goes about proving such a claim. – Ashvin Swaminathan Jul 26 '15 at 23:39
• just work with $X_n/n$... – ofer zeitouni Aug 1 '15 at 12:52
• The trouble with using $X_n/n$ is that the only results I know about convergence of random variables hold for integer-valued random variables. Do analogous theorems hold for the renormalized variables? – Ashvin Swaminathan Aug 2 '15 at 20:39