Continuous mapping theorem: Difference between revisions

Revision as of 19:04, 30 March 2019

Revision as of 19:06, 30 March 2019

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===Convergence in distribution===

We will need a particular statement from the [[portmanteau theorem]]: that convergence in distribution <math>X_n\xrightarrow{d}X</math> is equivalent to

: <math>\limsup_{n\to\infty}\operatorname{Pr}(X_n \in F) \leq \operatorname{Pr}(X\in F) \text{ for every closed set } F.</math>

Fix an arbitrary closed set ''F''⊂''S′''. Denote by ''g''⁻¹(''F'') the pre-image of ''F'' under the mapping ''g'': the set of all points ''x''&nbsp;∈&nbsp;''S'' such that ''g''(''x'')∈''F''. Consider a sequence {''x_k''} such that ''g''(''x_k'')&nbsp;∈&nbsp;''F'' and ''x_k''&nbsp;→&nbsp;''x''. Then this sequence lies in ''g''⁻¹(''F''), and its limit point ''x'' belongs to the [[closure (topology)|closure]] of this set, <span style="text-decoration:overline">''g''⁻¹(''F'')</span> (by definition of the closure). The point ''x'' may be either:

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Consider the event {''g''(''X_n'')∈''F''}. The probability of this event can be estimated as

: <math>

\operatorname{Pr}\big(g(X_n)\in F\big) = \operatorname{Pr}\big(X_n\in g^{-1}(F)\big) \leq \operatorname{Pr}\big(X_n\in \overline{g^{-1}(F)}\big),

</math>

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) =

\operatorname{Pr}\big(g(X) \in F\big) + 0.

\end{align}</math>