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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
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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.
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