distributions of a stochastic process
Just as one can associate a random variable with its distribution , one can associate a stochastic process with some distributions, such that the distributions will more or less describe the process. While the set of distributions can describe the random variables individually, it says nothing about the relationships between any pair, or more generally, any finite set of random variables ’s at different ’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 ’s, called the family of finite dimensional probability distributions. Specifically, let be any positive integer, an -dimensional probability distribution of the stochastic process is a joint probability distribution of , where :
The set of all -dimensional probability distributions for each and each set of is called the family of finite dimensional probability distributions, or family of finite dimensional distributions, abbreviated f.f.d., of the stochastic process .
Let be a permutation on . For any and , define and . Then
We say that the finite probability distributions are consistent with one another if, for any , each set of ,
Two stochastic processes and are said to be identically distributed, or versions of each other if
and have the same f.f.d.
|Title||distributions of a stochastic process|
|Date of creation||2013-03-22 15:21:35|
|Last modified on||2013-03-22 15:21:35|
|Last modified by||CWoo (3771)|
|Synonym||finite dimensional probability distributions|
|Defines||finite dimensional distributions|
|Defines||identically distributed stochastic processes|
|Defines||version of a stochastic process|