Editing Independence (probability theory) (section)

====For one stochastic process====
The definition of independence may be extended from random vectors to a [[stochastic process]]. Therefore, it is required for an independent stochastic process that the random variables obtained by sampling the process at any <math>n</math> times <math>t_1,\ldots,t_n</math> are independent random variables for any <math>n</math>.<ref name=HweiHsu>{{cite book| last1=Hwei| first1=Piao| title=Theory and Problems of Probability, Random Variables, and Random Processes| publisher=McGraw-Hill| year=1997| isbn=0-07-030644-3| url-access=registration| url=https://archive.org/details/schaumsoutlineof00hsuh}}</ref>{{rp|p. 163}}

Formally, a stochastic process <math>\left\{ X_t \right\}_{t\in\mathcal{T}}</math> is called independent, if and only if for all <math>n\in \mathbb{N}</math> and for all <math>t_1,\ldots,t_n\in\mathcal{T}</math>

{{Equation box 1
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|equation = {{NumBlk||<math>F_{X_{t_1},\ldots,X_{t_n}}(x_1,\ldots,x_n) = F_{X_{t_1}}(x_1) \cdot \ldots \cdot F_{X_{t_n}}(x_n) \quad \text{for all } x_1,\ldots,x_n</math>|{{EquationRef|Eq.7}}}}
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where {{nowrap|<math>F_{X_{t_1},\ldots,X_{t_n}}(x_1,\ldots,x_n) = \mathrm{P}(X(t_1) \leq x_1,\ldots,X(t_n) \leq x_n)</math>.}} Independence of a stochastic process is a property ''within'' a stochastic process, not between two stochastic processes.