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Suppose that $X_n$ are random variables that converge to $X$ in distribution and $f$ is a continuous function prove that $f(X_n)$ also converge to $f(X)$ in distribution

I have no idea which theorem or kind of method I should use. Thanks!

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That $(X_n)$ converges to $X$ in distribution means that $$ E[g(X_n)]\to E[g(X)]\quad n\to\infty $$ for every bounded and continuous function $g:\mathbb{R}\to\mathbb{R}$. In order to show that $(f(X_n))$ converges in distribution to $f(X)$, you want to show that $$ E[g(f(X_n))]\to E[g(f(X))]\quad n\to\infty $$ for every bounded and continuous function $g:\mathbb{R}\to\mathbb{R}$. But $$ E[g(f(X_n))]=E[(g\circ f)(X_n)] $$ and what can we say about $g\circ f$ given the assumptions on $g$ and $f$?

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Oh~Thanks a lot! – zyh-nankai university Dec 13 '12 at 17:24

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