= Factorial-moment criterion for Poisson convergence
{title2=$\mathbb E(X_n)_k\to\lambda^k$}
If nonnegative integer-valued <random variables> $X_n$ have all fixed <factorial moments> converging to $\lambda^k$, then their <convergence in distribution> is to the <Poisson distribution> with mean $\lambda$. The first-moment bound gives <uniform tightness>, and bounds on higher moments give <uniform integrability> of each fixed lower moment. Every subsequential limit therefore has the indicated <factorial moments>. For such a limit $X$ and $t\geq0$, the nonnegative binomial expansion and the <monotone convergence theorem> give $\mathbb E(1+t)^X=\sum_{k\geq0}t^k\lambda^k/k!=e^{\lambda t}$. This identifies its <moment-generating function> as that of the <Poisson distribution>, uniquely determining the limit.
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