Continuous mapping theorem: Difference between revisions

Revision as of 19:42, 30 March 2019

Revision as of 19:43, 30 March 2019

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at each point ''X''(''ω'') where ''g''(·) is continuous. Therefore,

: <math>\begin{align}

\Pr\Bigleft(\lim_{n\to\infty}g(X_n) = g(X)\Bigright)

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

&\geq \Pr\Bigleft(\lim_{n\to\infty}X_n = X,\ X\notin D_g\Bigright) = 1,

\end{align}</math>

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