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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>{{harvtxtcite 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>{{harvnbcite 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==
Let {''X<sub>n</sub>''}, ''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<sub>g</sub>'' such that {{nowrap|1=Pr[''X'' ∈ ''D<sub>g</sub>''] = 0}}. Then<ref>{{harvnbcite book |Van derlast = Billingsley Vaart|1998 first = Patrick |loc authorlink =Theorem 2.3,Patrick pageBillingsley 7}}</ref><ref>{{harvnb|Billingsley title = Convergence of Probability Measures | year = 1969 | publisher = John Wiley & Sons | isbn = 0-471-07242-7| ref = harv |page=31, (Corollary 1) }}</ref><ref>{{harvnbcite book |Billingsley last = Van der Vaart |1999 first = A. W. | title = Asymptotic Statistics | year = 1998 | publisher = Cambridge University Press | ___location = New York | isbn = 0-521-49603-9 | url =https://books.google.com/books?id=UEuQEM5RjWgC&pg=PA7 | ref = CITEREFVan_der_Vaart1998 |page=21,7 (Theorem 2.73) }}</ref>
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==References==
{{reflist}}
==Further reading==
* {{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
| ref = harv
}}
* {{cite book
| last = Billingsley
| first = Patrick
| authorlink = Patrick Billingsley
| title = Convergence of Probability Measures
| year = 1969
| publisher = John Wiley & Sons
| isbn = 0-471-07242-7| ref = harv
}}
* {{cite book
| last = Billingsley
| first = Patrick
| title = Convergence of Probability Measures
| year = 1999
| publisher = John Wiley & Sons
| edition = 2nd
| isbn = 0-471-19745-9
| url = https://books.google.com/books?id=QY06uAAACAAJ
| ref = harv
}}
* {{cite journal
| doi = 10.1214/aoms/1177731415
| last = Mann |first=H. B.
| authorlink = Henry Mann
| last2=Wald |first2=A.
| authorlink2 = Abraham Wald
| 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
}}
* {{cite book
| last = Van der Vaart
| first = A. W.
| title = Asymptotic statistics
| year = 1998
| publisher = Cambridge University Press
| ___location = New York
| isbn = 0-521-49603-9
| url = https://books.google.com/books?id=UEuQEM5RjWgC
| ref = CITEREFVan_der_Vaart1998
}}
[[Category:Probability theorems]]
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