Content deleted Content added
m →Proof: reference |
→Convergence almost surely: + proof |
||
Line 32:
===Convergence almost surely===
By definition of the continuity of function ''g(·)'',
: <math>
\lim_{n\to\infty}X_n(\omega) = X(\omega) \quad\Rightarrow\quad \lim_{n\to\infty}g(X_n(\omega)) = g(X(\omega))
</math>
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_n\notin D_g\Big) \geq \\
& \operatorname{Pr}\Big(\lim_{n\to\infty}X_n = X,\ X\notin D_g\Big) \geq
\operatorname{Pr}\Big(\lim_{n\to\infty}X_n = X\Big) - \operatorname{Pr}(X\in D_g) = 1-0 = 1.
\end{align}</math>
By definition, we conclude that ''g(X<sub>n</sub>)'' converges to ''g(X)'' almost surely.
==References==
| |||