Independent Random Elements: Preimages, Functions and Blocks

In the setting of Probability Spaces: Notation, let $J$ be a set.

(i) Independence tested on preimages. Let $((S_{j},d_{j}))_{j\in J}$ be a family of metric spaces and let $(X_{j})_{j\in J}$ be a family in which each $X_{j}$ is a random element of $S_{j}$. Then $(X_{j})_{j\in J}$ is independent if and only if for every $n\in\mathbb{N}$, all distinct $j_{1},\dots,j_{n}\in J$ and all $B_{1}\in\mathcal{B}(S_{j_{1}}),\dots,B_{n}\in\mathcal{B}(S_{j_{n}})$,

$$P(X_{j_{1}}\in B_{1},\dots,X_{j_{n}}\in B_{n})=\prod_{k=1}^{n}P(X_{j_{k}}\in B_{k}).$$

(ii) Independence of two random elements tested on preimages. Let $X$ and $Y$ be random elements of metric spaces $(S,d)$ and $(T,e)$. Then $X$ and $Y$ are independent if and only if

$$P(X\in B,\ Y\in B')=P(X\in B)\,P(Y\in B')\qquad(B\in\mathcal{B}(S),\ B'\in\mathcal{B}(T)).$$

(iii) Measurable functions preserve independence. Let $((S_{j},d_{j}))_{j\in J}$ and $((T_{j},e_{j}))_{j\in J}$ be families of metric spaces, let $(X_{j})_{j\in J}$ be a family in which each $X_{j}$ is a random element of $S_{j}$, and let $(g_{j})_{j\in J}$ be a family of functions $g_{j}:S_{j}\to T_{j}$, each $\mathcal{B}(S_{j})/\mathcal{B}(T_{j})$-measurable. If $(X_{j})_{j\in J}$ is independent, then so is the family $(g_{j}(X_{j}))_{j\in J}$ of random elements of the $T_{j}$ (Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii)).

(iv) Measurable functions of an independent pair. Let $X$ and $Y$ be random elements of metric spaces $(S,d)$ and $(T,e)$, let $(S',d')$ and $(T',e')$ be metric spaces, and let $g:S\to S'$ and $h:T\to T'$ be $\mathcal{B}(S)/\mathcal{B}(S')$- and $\mathcal{B}(T)/\mathcal{B}(T')$-measurable. If $X$ and $Y$ are independent, then so are $g(X)$ and $h(Y)$.

(v) Disjoint blocks. Let $m,n\in\mathbb{N}$ with $m<n$, and let $(X_{k})_{k\in[n]}$ be an independent family of random variables. Let $U:\Omega\to\mathbb{R}^{m}$ and $V:\Omega\to\mathbb{R}^{n-m}$ be the functions

$$U(\omega)=(X_{1}(\omega),\dots,X_{m}(\omega)),\qquad V(\omega)=(X_{m+1}(\omega),\dots,X_{n}(\omega))\qquad(\omega\in\Omega)$$

(Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (iii)), which are random vectors by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (i). Then $U$ and $V$ are independent.