distributions of a stochastic process

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

)

∣

t

∈

T

}

with some distributions, such that the distributions will more or less describe the process. While the set of distributions

{

F

X

⁢

(

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

⁢

(

t

1

)

,

…

,

X

⁢

(

t

n

)

, where

t

i

∈

T

:

F

t

1

,

…

,

t

n

(

x

1

,

…

,

x

n

)

:=

F

X

⁢

(

t

1

)

,

…

,

X

⁢

(

t

n

)

(

x

1

,

…

,

x

n

)

=

P

(

{

X

(

t

1

)

≤

x

1

}

∩

⋯

∩

{

X

(

t

n

)

≤

x

n

}

)

.

The set of

all

n

-dimensional probability distributions for each

n

∈

ℤ

+

and each set of

t

1

,

…

,

t

n

∈

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

t

1

,

…

,

t

n

∈

T

and

x

1

,

…

,

x

n

∈

ℝ

, define

s

i

=

t

σ

⁢

(

i

)

and

y

i

=

x

σ

⁢

(

i

)

. Then

F

s

1

,

…

,

s

n

⁢

(

y

1

,

…

,

y

n

)

=

P

(

{

X

(

s

1

)

≤

y

1

}

∩

⋯

∩

{

X

(

s

n

)

≤

y

n

}

)

=

P

(

{

X

(

t

1

)

≤

x

1

}

∩

⋯

∩

{

X

(

t

n

)

≤

x

n

}

)

=

F

t

1

,

…

,

t

n

⁢

(

x

1

,

…

,

x

n

)

.

We say that the finite probability distributions are

consistent

with one another if, for any

n

, each set of

t

1

,

…

,

t

n

+

1

∈

T

,

F

t

1

,

…

,

t

n

⁢

(

x

1

,

…

,

x

n

)

=

lim

x

n

+

1

→

∞

⁡

F

t

1

,

…

,

t

n

,

t

n

+

1

⁢

(

x

1

,

…

,

x

n

,

x

n

+

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.

Title

distributions of a stochastic process

Canonical

name

DistributionsOfAStochasticProcess

Date of creation

2013-03-22 15:21:35

Last modified on

2013-03-22 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

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