0.0.1 Continuous Mapping Theorem
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables.
Let \(\{ X_{n} \}\) be a sequence of random variables and \(g\) some continuous function. Then - \(X_n\stackrel{\mathcal{D}}{\to}X_n\implies g(X_n)\stackrel{\mathcal{D}}{\to}g(X)\) - \(X_n\stackrel{\mathbb{P}}{\to}X_n\implies g(X_n)\stackrel{\mathbb{P}}{\to}g(X)\) - \(X_n\stackrel{a.s.}{\to}X_n\implies g(X_n)\stackrel{a.s.}{\to}g(X)\)
The intuition behind the result is clear, since Heine’s definition of a continuous function is one that maps convergent sequences onto convergent sequences. ### Continuous Mapping Theorem (Detailed)
Let \(\{ X_{n} \}\) be a sequence of r.v.s such that \(X_{n}\stackrel{\mathcal{D}}{\to}X\). Let \(g:\mathbb{R}\to \mathbb{R}\) be a function with discontinuity set \(D(g):=(C(g))^c=\{ x:g\text{ is not continuous at }x \}\). Let \(\mathbb{P}(X \in D(g))=0.\) Then \(g(X_{n})\stackrel{\mathcal{D}}{\to} g(X)\).
If, in addition, \(g\) is bounded, the dominated convergence theorem implies that \[
\int g(x) dF_{n}(x)\to \int g(x)dF(x)\implies \mathbb{E}[g(X_{n})]\to \mathbb{E}[g(X)].
\]