Skorokhod's representation theorem explained
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space. It is named for the Ukrainian mathematician A. V. Skorokhod.
Statement
Let
be a sequence of probability measures on a
metric space
such that
converges weakly to some probability measure
on
as
. Suppose also that the
support of
is
separable. Then there exist
-valued random variables
defined on a common probability space
such that the law of
is
for all
(including
) and such that
converges to
,
-almost surely.
See also
- Convergence in distribution
References
- Book: Billingsley, Patrick . Convergence of Probability Measures . registration . John Wiley & Sons, Inc. . New York . 1999 . 0-471-19745-9. (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)
