Editing Stochastic process (section)

====Finite-dimensional probability distributions====
{{Main|Finite-dimensional distribution}}
For a stochastic process <math>X</math> with law <math>\mu</math>, its '''finite-dimensional distribution''' for <math>t_1,\dots,t_n\in T</math> is defined as:
<div class="center"><math>
\mu_{t_1,\dots,t_n} =P\circ (X({t_1}),\dots, X({t_n}))^{-1},
</math></div>
This measure <math>\mu_{t_1,..,t_n}</math> is the joint distribution of the random vector <math>
(X({t_1}),\dots, X({t_n}))
</math>; it can be viewed as a "projection" of the law <math>\mu</math> onto a finite subset of <math>T</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page123">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=123}}</ref>

For any measurable subset <math>C</math> of the <math>n</math>-fold [[Cartesian power]] <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as:<ref name="Lamperti1977page1"/>
<div class="center"><math>
\mu_{t_1,\dots,t_n}(C) =P \Big(\big\{\omega\in \Omega: \big( X_{t_1}(\omega), \dots, X_{t_n}(\omega) \big) \in C \big\} \Big).
</math></div>
The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.<ref name="Rosenthal2006page177"/>