Content deleted Content added
←Redirected page to Slutsky's theorem |
content split out from the Slutsky’s theorem article |
||
Line 1:
If <math> X_n </math> are [[random element]]s with values in a [[metric space]] and
<math> X_n \xrightarrow{\mathcal D} \, X</math>, <math>h</math> is a function on the metric space, and the probability that <math> X </math> attains a value where <math>h</math> is [[discontinuous]] is zero, then <math>h(X_n)\xrightarrow{\mathcal D}h(X)</math> (<ref name="Billingsley-1969-CPM">{{cite book
| last = Billingsley
| first = Patrick
| title = Convergence of Probability Measures
| year = 1969
| publisher = John Wiley & Sons}} ISBN 0471072427
</ref> page 31, Corollary 1,
<ref>{{cite book
| last = Billingsley
| first = Patrick
| title = Convergence of Probability Measures
| year = 1999
| publisher = John Wiley & Sons
| page = 2nd edition}} ISBN 0471197459
</ref> page 21, Theorem 2.7)
This includes for example the convergence of the sum of two sequences of random variables <math> X_n </math> and <math> Y_n </math> (the random element is the pair of the random variables, the continuous function is the mapping of the pair to the result of the operation), but only in the case where
::<math>(X_n, Y_n) \, \xrightarrow{\mathcal D} \, (X,Y).</math>
We note that this does not lead to a more general case of Slutsky's Theorem, because that would require only the assumption
::<math>X_n \, \xrightarrow{\mathcal D} \, X</math> and <math>Y_n \, \xrightarrow{\mathcal D} \, Y,</math>
which does not imply <math>(X_n, Y_n) \, \xrightarrow{\mathcal D} \, (X,Y)</math>, so we cannot apply the Continuous mapping theorem.
==References==
{{refbegin}}
{{refend}}
===Notes===
{{refs}}
| |||