Continuous mapping theorem: Difference between revisions

Revision as of 19:55, 10 September 2011

Revision as of 17:13, 13 September 2011

Line 4:

==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''&thinsp;∈&thinsp;''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>X_n \ \xrightarrow{p}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{p}\ g(X);</math>