ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Simple map
Simple function
Measurable simple complex function
Random simple number
Random Boolean number
Bernoulli random boolean number
Poisson random natural number
Definition D5524
Standard Poisson random natural number
Formulation 0
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}