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