Continuous mapping theorem: Difference between revisions

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In [[probability theory]], the '''continuous mapping theorem''' states that continuous functions retain their [[Continuous_function#Heine_definition_of_continuity|limit-preserving]] properties in the sense of the [[convergence of random variables]]. A continuous function, in [[Continuous_function#Heine_definition_of_continuity|Heine’s definition]], is such a function which maps convergent sequences into convergent sequences: if ''x_n'' → ''x'' then ''g''(''x_n)'') → ''g''(''x)''). The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {''x_n''} with a sequence of random variables {''X_n''}, and replace the standard notion of convergence of real numbers “→” with one of the types of [[convergence of random variables]].

This theorem was first proved by {{harv|Mann|Wald|1943}}, and it is therefore sometimes is called the '''Mann–Wald theorem'''.<ref>{{harvnb|Amemiya|1985|page=88}}</ref>

==Statement==