Continuous mapping theorem: Difference between revisions

Revision as of 00:51, 14 September 2009

Revision as of 18:17, 14 September 2009

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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

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

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

&\geq \operatorname{Pr}\Big(\lim_{n\to\infty}X_n = X\Big) - \operatorname{Pr}(X\in D_g) = 1-0 = 1.

\end{align}</math>