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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 [[Henry Mann]] and [[Abraham Wald]] in 1943,<ref>{{cite journal | doi = 10.1214/aoms/1177731415 | last = Mann |first=H. B. | last2=Wald |first2=A. | year = 1943 | title = On Stochastic Limit and Order Relationships | journal = [[Annals of Mathematical Statistics]] | volume = 14 | issue = 3 | pages = 217–226 | jstor = 2235800 | ref = CITEREFMannWald1943 }}</ref> and it is therefore sometimes called the '''Mann–Wald theorem'''.<ref>{{cite book | last = Amemiya | first = Takeshi | authorlink = Takeshi Amemiya | year = 1985 | title = Advanced Econometrics | publisher = Harvard University Press | ___location = Cambridge, MA | isbn = 0-674-00560-0 | url = https://books.google.com/books?id=0bzGQE14CwEC&pg=pA88 | ref = harv |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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===Convergence in distribution===
We will need a particular statement from the [[portmanteau theorem]]: that convergence in distribution <math>X_n\xrightarrow{d}X</math> is equivalent to
: <math> \mathbb E f(X_n) \to \mathbb E f(X)</math> for every bounded continuous functional ''f''.
So it suffices to prove that <math> \mathbb E f(g(X_n)) \to \mathbb E f(g(X))</math> for every bounded continuous functional ''f''. Note that <math> F = f \circ g</math> is itself a bounded continuous functional. And so the claim follows from the statement above.
===Convergence in probability===
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