ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D2713
Independent random collection

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 $j \in J$
Let $\mathcal{P}_{\text{finite}}(J)$ be the D2337: Set of finite subsets of $J$.
The D1721: Random collection $X = \{ X_j \}_{j \in J}$ is independent if and only if $$\forall \, I \in \mathcal{P}_{\text{finite}}(J) \left[ \forall \, i \in I : E_i \in \sigma_{\text{pullback}} \langle X_i \rangle \quad \implies \quad \mathbb{P} \left( \bigcap_{i \in I} E_i \right) = \prod_{i \in I} \mathbb{P}(E_i) \right]$$

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 $$\forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \text{ distinct } j_1, \ldots, j_N \in J \left[ E_{j_1} \in \sigma_{\text{pullback}} \langle X_{j_1} \rangle, \ldots, E_{j_N} \in \sigma_{\text{pullback}} \langle X_{j_N} \rangle \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \right) = \prod_{n = 1}^N \mathbb{P}(E_{j_n}) \right]$$

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $X_j$ is a D202: Random variable 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 $$\forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \text{ distinct } j_1, \ldots, j_N \in J \left[ \{ X_{j_1} \in E_{j_1} \}, \ldots, \{ X_{j_N} \in E_{j_N} \} \in \mathcal{F} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N \{ X_{j_n} \in E_{j_n} \} \right) = \prod_{n = 1}^N \mathbb{P}(X_{j_n} \in E_{j_n}) \right]$$
Children
 ▶ I.I.D. random collection
Results
 ▶ Affine transformations preserve independent real pairs ▶ Measurable transformation preserves independent countable random collection ▶ Measurable transformation preserves independent finite random collection ▶ Uncorrelated random collection need not be independent