independent stochastic processes

Two stochastic processes $\{X(t)\mid t\in T\}$ and $\{Y(t)\mid t\in T\}$ are said to be if for any positive integer $n<\infty$ , and any sequence $t_{1},\ldots,t_{n}\in T$ , the random vectors $\boldsymbol{X}:=(X(t_{1}),\ldots,X(t_{n}))$ and $\boldsymbol{Y}:=(Y(t_{1}),\ldots,Y(t_{n}))$ are independent. This means, for any two $n$ -dimensional Borel sets $A,B\subseteq\mathbb{R}^{n}$ , we have

$P\Big{[}\boldsymbol{X}^{-1}(A)\cap\boldsymbol{Y}^{-1}(B)\Big{]}=P\Big{[}% \boldsymbol{X}^{-1}(A)\Big{]}P\Big{[}\boldsymbol{Y}^{-1}(B)\Big{]}.$

Title	independent stochastic processes
Canonical name	IndependentStochasticProcesses
Date of creation	2013-03-22 15:24:36
Last modified on	2013-03-22 15:24:36
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60G07