Precompact Stochastic Process
Theorem
Let’s define a function space consisting of continuous functions going from measurable space $(S, \mathcal{S})$ to $(S ', \mathcal{S} ')$ as $\mathscr{H}:= C \left( S,S’ \right)$, and say that $\left\{ h^{-1}(A’): h \in \mathscr{H} , A ' \in \mathcal{S} ' \right\}$ is a separating class of $(S , \mathcal{S})$. Here, $X$ is a probability element defined in $S$, and $\left\{ X_n \right\}_{n \in \mathbb{N}}$ is a stochastic process defined in $S$.
If
- (i) $\left\{ X_{n} \right\}$ is pre-compact.
- (ii) For all $h \in \mathscr{H}$, $h \left( X_{n} \right) \overset{D}{\to} h(X)$
then, $X_{n} \overset{D}{\to} X$ holds.
Explanation
The continuous mapping theorem required the condition $P(X \in C_{h})=1$ to demonstrate that $X_{n} \overset{D}{\to} X \implies h(X_{n}) \overset{D}{\to} h(X)$ is true, likewise, to prove its converse, the condition of being pre-compact is necessary.
Proof
Under assumption (i), that the stochastic process $\left\{ X_{n} \right\}$ is pre-compact means that for every subsequence $\left\{ X_{n '} \right\} \subset \left\{ X_{n} \right\}$, there exists a further subsequence $$\left\{ X_{n ''} \right\} \subset\left\{ X_{n '} \right\} \subset \left\{ X_{n} \right\}$$ that converges to some $Y \in S$. In other words, $Y \in S$ and $\left\{ X_{n ''} \right\}$ satisfying $X_{n ''} \overset{D}{\to} Y$ exist, and then for all $h \in \mathscr{H}$, $$h \left( X_{n ''} \right) \overset{D}{\to} h \left( Y \right)$$ and as per assumption (ii), $h \left( X_{n} \right) \overset{D}{\to} h(X)$, so $$h \left( X_{n ''} \right) \overset{D}{\to} h \left( X \right)$$ Meanwhile, since $\left\{ h^{-1}(A’): h \in \mathscr{H} , A ' \in \mathcal{S} ' \right\}$ is placed as a separating class, for all $A \in \mathcal{S} ' $ and $h \in \mathscr{H}$, $$ \begin{align*} & h(X) \overset{D}{=} h(Y) \\ \iff & P \left( X \in h^{-1}(A) \right) = P \left( Y \in h^{-1}(A) \right) \\ \iff & P \circ X^{-1} = P \circ Y^{-1} \qquad \text{, on }\left\{ h^{-1}(A’): h \in \mathscr{H} , A ' \in \mathcal{S} ' \right\} \\ \color{red}{\iff}& P \circ X^{-1} = P \circ Y^{-1} \qquad \text{, on } (S,\mathcal{S}) \\ \iff &X \overset{D}{=} Y \end{align*} $$
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