e consider several random variables
given by the joint
distribution
For every variable
we
introduce
Suppose
is some uniform random variable supported on
.
Observe
that
Hence, the variable
is distributed as
for some
,
uniformly distributed on [0,1].
Therefore,

Summary

(Sklar theorem 1) Let
be uniform on
random variables given by joint distribution
.
Pick a set of non decreasing functions
and introduce the random variables
then

(Sklar theorem 1)

Summary

(Sklar theorem 2) Conversely, for any set of random variables
given by the joint distribution
the
expression

(Sklar theorem 2)

is a joint distribution for some uniform on
random variables
.