Continuous mapping theorem: Difference between revisions

Revision as of 13:21, 7 December 2022

Latest revision as of 06:32, 14 April 2025

(5 intermediate revisions by 4 users not shown)

Line 1:

{{Short description|Probability theorem}}

{{Distinguish|text=the [[contraction mapping theorem]]}}

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’sHeine's definition]], is such a function that 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 [[Henry Mann]] and [[Abraham Wald]] in 1943,<ref>{{cite journal | doi = 10.1214/aoms/1177731415 | last1 = Mann |first1=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 | doi-access = free }}</ref> and it is therefore sometimes called the '''Mann–Wald theorem'''.<ref>{{cite book | last = Amemiya | first = Takeshi | author-link = 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 |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_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'' ∈ ''D_g''] = 0}}. Then<ref>{{cite book | last = Billingsley | first = Patrick | author-link = Patrick Billingsley | title = Convergence of Probability Measures | year = 1969 | publisher = John Wiley & Sons | isbn = 0-471-07242-7|page=31 (Corollary 1) }}</ref><ref>{{cite book | last = Vanvan 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&pg=PA7 |page=7 (Theorem 2.3) }}</ref>

: <math>

Line 17 ⟶ 18:

==Proof==

<div style="NO-align:right"><small>This proof has been adopted from {{harv|Vanvan der Vaart|1998|loc=Theorem 2.3}}</small></div>

Spaces ''S'' and ''S′'' are equipped with certain metrics. For simplicity we will denote both of these metrics using the |''x''&nbsp;−&nbsp;''y''| notation, even though the metrics may be arbitrary and not necessarily Euclidean.

Line 25 ⟶ 26:

: <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''. For simplicity we assume ''g'' continuous. Note that <math> F = f \circ g</math> is itself a bounded continuous functional. And so the claim follows from the statement above. The general case is slightly more technical.

===Convergence in probability===

Line 69 ⟶ 70:

{{reflist}}

[[Category:ProbabilityTheorems theoremsin probability theory]]

[[Category:Theorems in statistics]]

[[Category:Articles containing proofs]]