Continuous mapping theorem: Difference between revisions

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==Statement==

Let {''X_n''}, ''X'' be [[random element]]s defined on a [[metric space]] ''S''. Suppose a function {{nowrap|''g'': ''S''→''S′''}} (where ''S′'' is another metric space) has the set of [[Discontinuity (mathematics)|discontinuity points]] ''D_g'' such that {{nowrap|1=Pr[''X'' ∈ ''D_g''] = 0}}. Then<ref>{{harvnb|Van der Vaart|1998|loc=Theorem 2.3, page 7}}</ref><ref>{{harvnb|Billingsley|1969|page=31, Corollary 1}}</ref><ref>{{harvnb|Billingsley|1999|page=21, Theorem 2.7}}</ref>

# <math>X_n \ \xrightarrow{d}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{d}\ g(X);</math>

: <math>

# <math>X_n \ \xrightarrow{p}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{p}\ g(X);</math>

\begin{align}

# <math>X_n \ \xrightarrow{\!\!as\!\!}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{\!\!as\!\!}\ g(X).</math>

# <math>X_n \ \xrightarrow\text{d}\ X \quad & \Rightarrow\quad g(X_n)\ \xrightarrow\text{d}\ g(X);</math> \\[6pt]

# <math>X_n \ \xrightarrow\text{p}\ X \quad & \Rightarrow\quad g(X_n)\ \xrightarrow\text{p}\ g(X);</math> \\[6pt]

# <math>X_n \ \xrightarrow\text{\!\!as\!\!a.s.}\ X \quad & \Rightarrow\quad g(X_n)\ \xrightarrow\text{\!\!as\!\!a.s.}\ g(X).</math>

\end{align}

</math>

where the superscripts, "d", "p", and "a.s." denote [[convergence in distribution]], [[convergence in probability]], and [[almost sure convergence]] respectively.

==Proof==