We recall definition of convergence in distribution and its related Portmanteau Theorem here. This is based on [Bil99] ([Patrick Billingsley 1999]).
Definition 1
is called random element if it satisfies the following conditions:
-
is a probability space;
-
for some metric space
.
Random elements are said to be convergent to an random element
in distribution, and denoted by
, if
– QED. –
The following theorem can be found in Page 26 of [Bil99].
Theorem 2 (Portmanteau Theorem)
The following five statements are equivalent:
-
;
-
for all bounded uniformly continuous
;
-
for all closed
;
-
for all open
;
-
for all
-continuity sets
.
– QED –
No. 3 of Pormanteau Theorem can be written by
which is indeed to be still true for all bounded upper semicontinuous functions in place of . Similarly, No. 4 is equivalent to
which can be extended to all bounded lower semicontinuous functions.
If , then
-
for all bounded upper semicontinuous
;
-
for all bounded lower semicontinuous
.
Proof: If is bounded upper semicontinuous, then there exists bounded continuous functions
such that
. Thus,
Note that, by bounded convergence theorem, and it yields the first conclusion. The second one is the consequence of the first conclusion applying to upper semicontinuous function
.
– QED –
We can also extend the above example even further to -almost semicontinuous functions.
Example 2
Let .
- If
is bounded upper semicontinuous
-almost surely, then
;
- If
is bounded lower semicontinuous
-almost surely, then
.
Proof: We will only show the first result. The condition
is upper semicontinuous
-almost surely
means that
such that
and
is USC in
.
Denote , and define
. Then
is bounded USC and this yields by Example 1
One can conclude the result by easily checking that