Stochastic process
Finite-dimensional probability distributions

For a stochastic process X {\displaystyle X} with law μ {\displaystyle \mu }, its finite-dimensional distribution for t 1, …, t n ∈ T {\displaystyle t_{1},\dots,t_{n}\in T} is defined as:

This measure μ t 1,.., t n {\displaystyle \mu _{t_{1},..,t_{n}}} is the joint distribution of the random vector (X (t 1), …, X (t n)) {\displaystyle (X({t_{1}}),\dots,X({t_{n}}))}; it can be viewed as a "projection" of the law μ {\displaystyle \mu } onto a finite subset of T {\displaystyle T}.

For any measurable subset C {\displaystyle C} of the n {\displaystyle n} -fold Cartesian power S n = S × ⋯ × S {\displaystyle S^{n}=S\times \dots \times S}, the finite-dimensional distributions of a stochastic process X {\displaystyle X} can be written as:

The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.
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