Continuous mapping theorem: Difference between revisions

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In [[probability theory]], the '''continuous mapping theorem''' states that continuous functions are [[Continuous function#Heine definition of continuity|limit-preserving]] 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_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 {{harvharvtxt|Mann|Wald|1943}}, and it is therefore sometimes called the '''Mann–Wald theorem'''.<ref>{{harvnb|Amemiya|1985|page=88}}</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''&thinsp;∈&thinsp; ∈ ''D_g''] = 0}}. Then<ref>{{harvnb|Van der Vaart|1998|loc=Theorem 2.3, page 7}}</ref><ref>{{harvnb|Billingsley|1969|page=31, Corollary 1}}</ref><ref>{{harvnb|Billingsley|1999|page=21, Theorem 2.7}}</ref>

# <math>X_n \ \xrightarrow{d}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{d}\ g(X);</math>

# <math>X_n \ \xrightarrow{p}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{p}\ g(X);</math>

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and by the portmanteau theorem the [[limsup]] of the last expression is less than or equal to Pr(''X''&nbsp;∈&nbsp;<span style="text-decoration:overline">''g''⁻¹(''F'')</span>). Using the formula we derived in the previous paragraph, this can be written as

: <math>\begin{align}

& \operatorname{Pr}\big(X\in \overline{g^{-1}(F)}\big) \leq

\operatorname{Pr}\big(X\in g^{-1}(F)\cup D_g\big) \leq \\

& \operatorname{Pr}\big(X \in g^{-1}(F)\big) + \operatorname{Pr}(X\in D_g) =

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Now suppose that |''g''(''X'')&nbsp;−&nbsp;''g''(''X_n'')|&nbsp;>&nbsp;''ε''. This implies that at least one of the following is true: either |''X''−''X_n''|&nbsp;≥&nbsp;''δ'', or ''X''&nbsp;∈&nbsp;''D_g'', or ''X''∈''B_δ''. In terms of probabilities this can be written as

: <math>

\Pr\big(\big|g(X_n)-g(X)\big|>\varepsilon\big) \leq

\Pr\big(|X_n-X|\geq\delta\big) + \Pr(X\in B_\delta) + \Pr(X\in D_g).

</math>

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===Convergence almost surely===

By definition of the continuity of the function ''g''(·),

: <math>

\lim_{n\to\infty}X_n(\omega) = X(\omega) \quad\Rightarrow\quad \lim_{n\to\infty}g(X_n(\omega)) = g(X(\omega))

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at each point ''X''(''ω'') where ''g''(·) is continuous. Therefore

: <math>\begin{align}

\operatorname{Pr}\Big(\lim_{n\to\infty}g(X_n) = g(X)\Big)

&\geq \operatorname{Pr}\Big(\lim_{n\to\infty}g(X_n) = g(X),\ X\notin D_g\Big) \\

&\geq \operatorname{Pr}\Big(\lim_{n\to\infty}X_n = X,\ X\notin D_g\Big) \\

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==References==

{{reflist}}

===Literature=Further reading==

{{refbegin}}

* {{cite book

| last = Amemiya

| first = Takeshi

| authorlink = Takeshi Amemiya

| year = 1985

| title = Advanced Econometrics

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| ___location = Cambridge, MA

| isbn = 0-674-00560-0

| url = https://books.google.com/books?id=0bzGQE14CwEC

| ref = harv

}}

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| last = Billingsley

| first = Patrick

| authorlink = Patrick Billingsley

| title = Convergence of Probability Measures

| year = 1969

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| publisher = John Wiley & Sons

| edition = 2nd

| isbn = 0-471-19745-9| ref = harv

| url = https://books.google.com/books?id=QY06uAAACAAJ

| ref = harv

}}

* {{cite journal

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| publisher = Cambridge University Press

| ___location = New York

| isbn = 978-0-521-49603-29

| url = https://books.google.com/books?id=UEuQEM5RjWgC

| ref = CITEREFVan_der_Vaart1998

}}

{{refend}}

===Notes===

{{reflist}}

[[Category:Probability theorems]]