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In [[probability theory]], the '''continuous mapping theorem''' states that continuous functions [[Continuous function#Heine definition of continuity|preserve limits]] even if their arguments are sequences of random variables. A continuous function, in [[Continuous function#Heine definition of continuity|Heine’s definition]], is such a function that maps convergent sequences into convergent sequences: if ''x<sub>n</sub>'' → ''x'' then ''g''(''x<sub>n</sub>'') → ''g''(''x''). The ''continuous mapping theorem'' states that this will also be true if we replace the deterministic sequence {''x<sub>n</sub>''} with a sequence of random variables {''X<sub>n</sub>''}, 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 {{harvtxt|Mann|Wald|1943}}, and it is therefore sometimes called the '''Mann–Wald theorem'''.<ref>{{harvnb|Amemiya|1985|page=88}}</ref> Meanwhile [[Denis Sargan]] refers to it as the '''general transformation theorem'''.<ref>{{cite book |first=Denis |last=Sargan |title=Lectures on Advanced Econometric Theory |___location=Oxford |publisher=Basil Blackwell |year=1988 |isbn=0-631-14956-2 |pages=4–8 }}</ref>
==Statement==
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