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distributions of a stochastic process∗ CWoo† 2013-03-21 19:22:29 Just as one can associate a random variable X with its distribution FX , one can associate a stochastic process {X(t) | t ∈ T } with some distributions, such that the distributions will more or less describe the process. While the set of distributions {FX(t) | t ∈ 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 n < ∞ be any positive integer, an n-dimensional probability distribution of the stochastic process {X(t) | t ∈ T } is a joint probability distribution of X(t1 ), . . . , X(tn ), where ti ∈ T : Ft1 ,...,tn (x1 , . . . , xn ) := FX(t1 ),...,X(tn ) (x1 , . . . , xn ) = P ({X(t1 ) ≤ x1 }∩· · ·∩{X(tn ) ≤ xn }). The set of all n-dimensional probability distributions for each n ∈ Z+ and each set of t1 , . . . , tn ∈ 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) | t ∈ T }. Let σ be a permutation on {1, . . . , n}. For any t1 , . . . , tn ∈ T and x1 , . . . , xn ∈ R, define si = tσ(i) and yi = xσ(i) . Then Fs1 ,...,sn (y1 , . . . , yn ) = P ({X(s1 ) ≤ y1 } ∩ · · · ∩ {X(sn ) ≤ yn }) = P ({X(t1 ) ≤ x1 } ∩ · · · ∩ {X(tn ) ≤ xn }) = Ft1 ,...,tn (x1 , . . . , xn ). ∗ hDistributionsOfAStochasticProcessi created: h2013-03-21i by: hCWooi version: h37183i Privacy setting: h1i hDefinitioni h60G07i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 We say that the finite probability distributions are consistent with one another if, for any n, each set of t1 , . . . , tn+1 ∈ T , Ft1 ,...,tn (x1 , . . . , xn ) = lim xn+1 →∞ Ft1 ,...,tn ,tn+1 (x1 , . . . , xn , xn+1 ). Two stochastic processes {X(t) | t ∈ T } and {Y (s) | s ∈ 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. 2