distributions of a stochastic process
Just as one can associate a random variable^{} $X$ with its distribution^{} ${F}_{X}$, one can associate a stochastic process^{} $\{X(t)\mid t\in T\}$ with some distributions, such that the distributions will more or less describe the process. While the set of distributions $\{{F}_{X(t)}\mid t\in T\}$ can describe the random variables $X(t)$ individually, it says nothing about the relationships between any pair, or more generally, any finite set^{} of random variables $X(t)$’s at different $t$’s. Another way is to look at the joint probability distribution of all the random variables in a stochastic process. This way we can derive the probability distribution functions of individual random variables. However, in most stochastic processes, there are infinitely many random variables involved, and we run into trouble right away.
To resolve this, we enlarge the above set of distribution functions to include all joint probability distributions of finitely many $X(t)$’s, called the family of finite dimensional probability distributions. Specifically, let $$ be any positive integer, an $n$dimensional probability distribution of the stochastic process $\{X(t)\mid t\in T\}$ is a joint probability distribution of $X({t}_{1}),\mathrm{\dots},X({t}_{n})$, where ${t}_{i}\in T$:
$${F}_{{t}_{1},\mathrm{\dots},{t}_{n}}({x}_{1},\mathrm{\dots},{x}_{n}):={F}_{X({t}_{1}),\mathrm{\dots},X({t}_{n})}({x}_{1},\mathrm{\dots},{x}_{n})=P(\{X({t}_{1})\le {x}_{1}\}\cap \mathrm{\cdots}\cap \{X({t}_{n})\le {x}_{n}\}).$$ 
The set of all $n$dimensional probability distributions for each $n\in {\mathbb{Z}}^{+}$ and each set of ${t}_{1},\mathrm{\dots},{t}_{n}\in T$ is called the family of finite dimensional probability distributions, or family of finite dimensional distributions, abbreviated f.f.d., of the stochastic process $\{X(t)\mid t\in T\}$.
Let $\sigma $ be a permutation on $\{1,\mathrm{\dots},n\}$. For any ${t}_{1},\mathrm{\dots},{t}_{n}\in T$ and ${x}_{1},\mathrm{\dots},{x}_{n}\in \mathbb{R}$, define ${s}_{i}={t}_{\sigma (i)}$ and ${y}_{i}={x}_{\sigma (i)}$. Then
${F}_{{s}_{1},\mathrm{\dots},{s}_{n}}({y}_{1},\mathrm{\dots},{y}_{n})$  $=$  $P(\{X({s}_{1})\le {y}_{1}\}\cap \mathrm{\cdots}\cap \{X({s}_{n})\le {y}_{n}\})$  
$=$  $P(\{X({t}_{1})\le {x}_{1}\}\cap \mathrm{\cdots}\cap \{X({t}_{n})\le {x}_{n}\})$  
$=$  ${F}_{{t}_{1},\mathrm{\dots},{t}_{n}}({x}_{1},\mathrm{\dots},{x}_{n}).$ 
We say that the finite probability distributions are consistent with one another if, for any $n$, each set of ${t}_{1},\mathrm{\dots},{t}_{n+1}\in T$,
$${F}_{{t}_{1},\mathrm{\dots},{t}_{n}}({x}_{1},\mathrm{\dots},{x}_{n})=\underset{{x}_{n+1}\to \mathrm{\infty}}{lim}{F}_{{t}_{1},\mathrm{\dots},{t}_{n},{t}_{n+1}}({x}_{1},\mathrm{\dots},{x}_{n},{x}_{n+1}).$$ 
Two stochastic processes $\{X(t)\mid t\in T\}$ and $\{Y(s)\mid s\in S\}$ are said to be identically distributed, or versions of each other if

1.
$S=T$, and

2.
$\{X(t)\}$ and $\{Y(s)\}$ have the same f.f.d.
Title  distributions of a stochastic process 
Canonical name  DistributionsOfAStochasticProcess 
Date of creation  20130322 15:21:35 
Last modified on  20130322 15:21:35 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60G07 
Synonym  finite dimensional probability distributions 
Synonym  ffd 
Related topic  StochasticProcess 
Related topic  KolmogorovsContinuityTheorem 
Related topic  ModificationOfAStochasticProcess 
Defines  finite dimensional distributions 
Defines  f.f.d. 
Defines  identically distributed stochastic processes 
Defines  version of a stochastic process 