We are going to show that, If in , and is continuous at almost surely, then . This is based on [Patrick Billingsley 1999] ([Bil99]).

Let be the space of RCLL -valued functions, equipped with Skorohod topology. is a given probability space.

**Proposition 1** * Let be random functions, and is convergent to in distribution (denoted by ). Let be a collection of at which is continuous almost surely. Then, for all . *

*Proof:* Fix . Let be a projection operator defined by . Note that, is continuous at if ([Bil99, P. 134]). Therefore, is continuous almost surely w.r.t . By mapping theorem ([Bil99, Thoerem 2.7]), implies .

Application of Proposition 1 is follows: for a continuous random function , to approximate a functional for appropriate functions and , it is enough to find in .

**Proposition 2** * Let be a continuous random function. If in , then *

*
** for all lower bounded continuous functions . *

*Proof:* First, we assume . By Fubini’s theorem and BCT, one can write LHS as

By Proposition 1, for all . Hence, it implies and . Once again, by using Fubini’s theorem, it leads to RHS. Secondly, we assume are non-negative. Define truncated function by for all and . Then, (1) holds for all , and Monotone convergence theorem implies (1) holds for non-negative continuous . Finally, we assume satisfy for some positive constant . Then, (1) holds for , and it implies (1) holds for .

### Like this:

Like Loading...