We give some definitions and basic properties of weak convergence of probability measures on metric spaces, based on [Bil99].
Let be a metric space.
are probability mesures on
. We write
for
measurable
for convenience.
Definition 1
is said to be weakly convergent to
, denoted by
, if
for every
.
The following theorem provides useful conditions equivalent to definition of weak convergence. A set is called
-continuity set, if
.
Theorem 2 (The Portmanteau Theorem, [Bil99, Theorem 2.1]) These five conditions are equivalent:
.
for all bounded uniformly continuous
.
for all closed
![]()
for all open
.
for all
-continuity sets
.
Let be the space of probability measures on
.
can be metrized by Prohov distance
([Bil99, P. 72]) if
is separable and complete metric space.
Theorem 3 ([Bil99, Theorem 6.8]) Suppose that
is separable and complete. Then
- weak convergence is equivalent to
-convergence,
is separable and complete
is relatively compact if and only if its
-closure is
-compact.
Let be another metric space, and
is measurable
. Then
is a probability measure on
.
Theorem 4 (Mapping theorem, [Bil99, Theorem 2.7]) Let
is measurable
, and
. In addition,
, where
be the set of discontinuities of
. Then
.
The above mapping theorem saids, weak convergence of probability measures are preserved by mapping , if
is continuous almost surely with respect to the limit measure
, i.e.
.
Let be a probability space, and
be measurable w.r.t.
. Then,
is called random element. The distribution of
is the probability measure
on
, which is sometimes denoted by
. If
is a measurable
, then
.
Definition 5 We say a sequence
converges in distribution to
, denoted by
, if
.
Thus, if and only if
for all
. We call a set
as an
-continuity set if
. Now, we can rewrite Theorem 2 in terms of random elements.
Theorem 6 ([Bil99, p. 26]) These five conditions are equivalent:
.
for all bounded uniformly continuous
.
for all closed
![]()
for all open
.
for all
-continuity sets
.
Also, the Theorem 4 can be rewritten into mapping theorem of random elements: on
implies
on
, if
is continuous almost surely with respect to
, i.e.