Let $\xi = \{ \{ \xi_{n, m} \}_{1 \leq m \leq n} \}_{n \geq 1}$ be a
D5164: Random real standard triangular array such that
| (i) |
\begin{equation}
\forall \, n \in 1, 2, 3, \ldots :
\forall \, m \in 1, \ldots, n :
\xi_{n, m} \overset{d}{=} \text{Bernoulli}(1 / n)
\end{equation}
|
| (ii) |
$\xi_{n, 1}, \ldots, \xi_{n, n}$ is an D2713: Independent random collection for each $n \geq 1$
|
A
D5216: Random natural number $X \in \text{Random}(\mathbb{N})$ is a
standard Poisson random natural number if and only if
\begin{equation}
X
\overset{d}{=} \lim_{n \to \infty} \sum_{m = 1}^n \xi_{n, m}
\end{equation}
