In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or finite collection of times).
Let
(
X
,
F
,
μ
)
be a measure space. The finite-dimensional distributions of
μ
are the pushforward measures
f
∗
(
μ
)
, where
f
:
X
→
R
k
,
k
∈
N
, is any measurable function.
Let
(
Ω
,
F
,
P
)
be a probability space and let
X
:
I
×
Ω
→
X
be a stochastic process. The finite-dimensional distributions of
X
are the push forward measures
P
i
1
…
i
k
X
on the product space
X
k
for
k
∈
N
defined by
P
i
1
…
i
k
X
(
S
)
:=
P
{
ω
∈
Ω
|
(
X
i
1
(
ω
)
,
…
,
X
i
k
(
ω
)
)
∈
S
}
.
Very often, this condition is stated in terms of measurable rectangles:
P
i
1
…
i
k
X
(
A
1
×
⋯
×
A
k
)
:=
P
{
ω
∈
Ω
|
X
i
j
(
ω
)
∈
A
j
f
o
r
1
≤
j
≤
k
}
.
The definition of the finite-dimensional distributions of a process
X
is related to the definition for a measure
μ
in the following way: recall that the law
L
X
of
X
is a measure on the collection
X
I
of all functions from
I
into
X
. In general, this is an infinite-dimensional space. The finite dimensional distributions of
X
are the push forward measures
f
∗
(
L
X
)
on the finite-dimensional product space
X
k
, where
f
:
X
I
→
X
k
:
σ
↦
(
σ
(
t
1
)
,
…
,
σ
(
t
k
)
)
is the natural "evaluate at times
t
1
,
…
,
t
k
" function.
It can be shown that if a sequence of probability measures
(
μ
n
)
n
=
1
∞
is tight and all the finite-dimensional distributions of the
μ
n
converge weakly to the corresponding finite-dimensional distributions of some probability measure
μ
, then
μ
n
converges weakly to
μ
.
