Let $\xi = \{ \{ \xi_{n, m} \}_{1 \leq m \leq n} \}_{n \geq 1}$ be a D5163: Random real triangular array such that

(i)	$\lambda \in (0, \infty)$ is a D5407: Positive real number
(ii)	$\theta_1, \theta_2, \theta_3, \ldots \in (0, 1]$ are each a D5407: Positive real number
(iii)	$$\begin{equation} \theta_n : = \min \left( \frac{\lambda}{n}, 1 \right) \end{equation}$$
(iv)	$$\begin{equation} \forall \, n \in 1, 2, 3, \ldots : \forall \, m \in 1, \ldots, n : \xi_{n, m} \overset{d}{=} \text{Bernoulli}(\theta_n) \end{equation}$$
(v)	$\xi_{n, 1}, \ldots, \xi_{n, n}$ is an D2713: Independent random collection for each $n \geq 1$

A D5216: Random natural number $X \in \text{Random}(\mathbb{N})$ is a Poisson random natural number with parameter $\lambda$ if and only if $$\begin{equation} X \overset{d}{=} \lim_{n \to \infty} \sum_{m = 1}^n \xi_{n, m} \end{equation}$$