Projection operator on space D

We are going to show that, If ${X_n \Rightarrow X}$ in ${\mathcal{D}^d}$, and ${X}$ is continuous at ${t}$ almost surely, then ${X_n(t) \Rightarrow X(t)}$. This is based on [Patrick Billingsley 1999] ([Bil99]).

Let ${\mathcal{D}^d = D([0,1]; \mathbb{R}^d)}$ be the space of RCLL ${\mathbb{R}^d}$-valued functions, equipped with Skorohod topology. ${(\Omega, \mathcal{F}, \mathbb{P})}$ is a given probability space.

Proposition 1 Let ${X_n, X: \Omega \rightarrow \mathcal{D}^d}$ be random functions, and ${X_n}$ is convergent to ${X}$ in distribution (denoted by ${X_n \Rightarrow X}$). Let ${C_X\subset [0,1]}$ be a collection of ${t}$ at which ${X}$ is continuous almost surely. Then, ${X_n(t) \Rightarrow X(t)}$ for all ${t\in C_X}$.

Proof: Fix ${t\in C_X}$. Let ${\pi_t: \mathcal{D}^d \rightarrow \mathbb{R}^d}$ be a projection operator defined by ${\pi_t(x) = x(t)}$. Note that, ${\pi_t}$ is continuous at ${x}$ if ${t \in C_x}$ ([Bil99, P. 134]). Therefore, ${\pi_t: \mathcal{D}^d \rightarrow \mathbb{R}^d}$ is continuous almost surely w.r.t ${\mathbb{P}X^{-1}}$. By mapping theorem ([Bil99, Thoerem 2.7]), ${X_n \Rightarrow X}$ implies ${X_n(t) \Rightarrow X(t)}$. $\Box$

Application of Proposition 1 is follows: for a continuous random function ${X}$, to approximate a functional ${\mathbb{E}[ \int_0^1 f(X(s)) ds + g(X(1))]}$ for appropriate functions ${f}$ and ${g}$, it is enough to find ${X_n \Rightarrow X}$ in ${\mathcal{D}^d}$.

Proposition 2 Let ${X: \Omega \rightarrow \mathcal{C}^d := C([0,1]; \mathbb{R}^d)}$ be a continuous random function. If ${X_n \Rightarrow X}$ in ${\mathcal{D}^d}$, then

$\displaystyle \lim_{n\rightarrow \infty}\mathbb{E}\Big[ \int_0^1 f(X_n(s)) ds + g(X_n(1))\Big] = \mathbb{E} \Big[ \int_0^1 f(X(s)) ds + g(X(1)) \Big] \ \ \ \ \ (1)$

for all lower bounded continuous functions ${f, g \in C(\mathbb{R}^d)}$.

Proof: First, we assume ${\|f\|_\infty + \|g\|_\infty < \infty}$. By Fubini’s theorem and BCT, one can write LHS as

$\displaystyle LHS = \int_0^1 \lim_{n\rightarrow \infty} \mathbb{E} [f(X_n(s))] ds + \lim_{n\rightarrow \infty} \mathbb{E}[g(X_n(1))].$

By Proposition 1, ${X_n(t) \Rightarrow X(t)}$ for all ${t\in [0,1]}$. Hence, it implies ${\lim_{n\rightarrow \infty} \mathbb{E} [f(X_n(s))] = f(X(s))}$ and ${\lim_{n\rightarrow \infty} \mathbb{E} [g(X_n(1))] = f(X(1))}$. Once again, by using Fubini’s theorem, it leads to RHS. Secondly, we assume ${f, g \in C(\mathbb{R}^d)}$ are non-negative. Define truncated function ${\varphi^k \in C_b(\mathbb{R}^d)}$ by ${ \varphi^k = \varphi \wedge k}$ for all ${\varphi \in C(\mathbb{R}^d)}$ and ${k \ge 0}$. Then, (1) holds for all ${(f^k, g^k)}$, and Monotone convergence theorem implies (1) holds for non-negative continuous ${(f, g)}$. Finally, we assume ${f, g \in C(\mathbb{R}^d)}$ satisfy ${f, g \ge -C}$ for some positive constant ${C}$. Then, (1) holds for ${(\hat f: = f + C, \hat g := g+C)}$, and it implies (1) holds for ${(f, g)}$. $\Box$