PlanetMath: independent sigma algebras

(more info)

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independent sigma algebras

(Definition)

Let $(\Omega, \mathcal{B}, P)$ be a probability space. Let $\mathcal{B}_1$ and $\mathcal{B}_2$ be two sub sigma algebras of $\mathcal{B}$ . Then $\mathcal{B}_1$ and $\mathcal{B}_2$ are said to be independent if for any pair of events $B_1\in\mathcal{B}_1$ and $B_2\in\mathcal{B}_2$ :

$\displaystyle P(B_1\cap B_2)=P(B_1)P(B_2).$

More generally, a finite set of sub- $\sigma$ -algebras $\mathcal{B}_1,\ldots, \mathcal{B}_n$ is independent if for any set of events $B_i\in \mathcal{B}_i$ , $i=1,\ldots,n$ :

$\displaystyle P(B_1\cap\cdots\cap B_n)=P(B_1)\cdots P(B_n).$

An arbitrary set $\mathcal{S}$ of sub- $\sigma$ -algebras is mutually independent if any finite subset of $\mathcal{S}$ is independent.

The above definitions are generalizations of the notions of independence for events and for random variables:

Events $B_1,\ldots,B_n$ (in $\Omega$ ) are mutually independent if the sigma algebras $\sigma(B_i):=\lbrace \varnothing, B_i, \Omega-B_i, \Omega\rbrace$ are mutually independent.
Random variables $X_1,\ldots,X_n$ defined on $\Omega$ are mutually independent if the sigma algebras $\mathcal{B}_{X_i}$ generated by the 's are mutually independent.

In general, mutual independence among events

, random variables

, and sigma algebras $\mathcal{B}_k$ means the mutual independence among $\sigma(B_i)$ , $\mathcal{B}_{X_j}$ , and $\mathcal{B}_k$ .

Remark. Even when random variables $X_1,\ldots, X_n$ are defined on different probability spaces $(\Omega_i,\mathcal{B}_i,P_i)$ , we may form the product of these spaces $(\Omega,\mathcal{B},P)$ so that (by abuse of notation) are now defined on $\Omega$ and their independence can be discussed.