Continuous Mapping Theorem

Definition

Continuous functions preserve convergences (in probability, Almost Surely, or in distribution) of a Sequence of random variables to limits.

Consider a Sequence $\{X_n\}$ of random variables defined on same Probability Space, and a Continuous Function $g$ on the space. Then,

• Convergence in Probability to a Constant: $X_n\stackrel{P}{\to }a\in \mathbb{R}⟹g(X_n)\stackrel{P}{\to }g(a)$
• Convergence in Probability: $X_n\stackrel{P}{\to }X⟹g(X_n)\stackrel{P}{\to }g(X)$
• Almost Sure Convergence: $X_n\stackrel{a.s.}{\to }X⟹g(X_n)\stackrel{a.s.}{\to }g(X)$
• Convergence in Distribution: $X_n\stackrel{D}{\to }X⟹g(X_n)\stackrel{D}{\to }g(X)$