ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D5524
Standard Poisson random natural number

Let $\xi = \{ \{ \xi_{n, m} \}_{1 \leq m \leq n} \}_{n \geq 1}$ be a D5164: Random real standard triangular array such that
 (i) $$\forall \, n \in 1, 2, 3, \ldots : \forall \, m \in 1, \ldots, n : \xi_{n, m} \overset{d}{=} \text{Bernoulli}(1 / n)$$ (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 $$X \overset{d}{=} \lim_{n \to \infty} \sum_{m = 1}^n \xi_{n, m}$$