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}
