Continuous mapping theorem: Difference between revisions

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In [[probability theory]], the '''continuous mapping theorem''' states that continuous functions are [[Continuous_functionContinuous function#Heine_definition_of_continuityHeine definition of continuity|limit-preserving]] even if their arguments are sequences of random variables. A continuous function, in [[Continuous_functionContinuous function#Heine_definition_of_continuityHeine 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 {{harv|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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B_\delta = \big\{x\in S\ \big|\ x\notin D_g:\ \exists y\in S:\ |x-y|<\delta,\, |g(x)-g(y)|>\varepsilon\big\}.

</math>

This is the set of continuity points ''x'' of the function ''g''(·) for which it is possible to find, within the ''δ''-neighborhood of ''x'', a point which maps outside the ''ε''-neighborhood of ''g''(''x''). By definition of continuity, this set shrinks as ''δ''  goes to zero, so that lim_''δ''→0''B_δ''  =  ∅.

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

: <math>

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

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</math>

On the right-hand side, the first term converges to zero as ''n''  →  ∞ for any fixed ''δ'', by the definition of convergence in probability of the sequence {''X_n''}. The second term converges to zero as ''δ''  →  0, since the set ''B_δ'' shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem. Therefore the conclusion is that

: <math>

\lim_{n\to\infty}\operatorname{Pr}\big(\big|g(X_n)-g(X)\big|>\varepsilon\big) = 0,

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==References==

===Literature===

{{refbegin}}