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, I am wondering whether it is still possible to show that the sequence $X_n$ of random variables weakly converges to a Poisson random variable. In particular, I would like to know whether there are any standard results that give the value of $$\lim_{n \to \infty} \mathbb{P}(X_n = \ell)$$ in this case. Any advice to this end is appreciated!

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!