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

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