Continuous mapping theorem: Difference between revisions

Revision as of 13:21, 7 December 2022

Revision as of 13:22, 7 December 2022

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==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>

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==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.