# 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 IndependentStochasticProcesses 2013-03-22 15:24:36 2013-03-22 15:24:36 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 60G07