We recall definition of convergence in distribution and its related Portmanteau Theorem here. This is based on [Bil99] ([Patrick Billingsley 1999]).
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.
- 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
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