ThmDex – An index of mathematical definitions, results, and conjectures.
F9075
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $X_j$ be a D202: Random variable on $P$ for each $j \in J$ such that
(i) $\sigma_{\text{pullback}} \langle X_j \rangle$ is the D1730: Pullback sigma-algebra of $X_j$ on $P$ for each $j \in J$
Then $X = \{ X_j \}_{j \in J}$ is an independent random collection on $P$ if and only if \begin{equation} \forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \text{ distinct } j_1, \ldots, j_N \in J \left[ E_{j_1} \in \sigma_{\text{pullback}} \langle X_{j_1} \rangle, \ldots, E_{j_N} \in \sigma_{\text{pullback}} \langle X_{j_N} \rangle \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \right) = \prod_{n = 1}^N \mathbb{P}(E_{j_n}) \right] \end{equation}