Continuous mapping theorem: Difference between revisions

Revision as of 19:19, 30 March 2019

Revision as of 19:41, 30 March 2019

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

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

\end{align}</math>,

because the intersection of two almost sure events is almost sure.