# Convergence of functions of random variables

I know that

IF $X_n$ is a sequence of random variables that converges to $X$ almost surely, then for any continuous function $g:\Bbb{R}\to\Bbb{R}\ ,\ g(X_n)$ converges to $g(X)$ almost surely.

Now if I replace almost surely with in probability, the statement still holds.

My question:

Does the statement hold for convergence in law?

• Yes; this is known as the continuous mapping or Mann-Wald theorem. – Math1000 Oct 1 '16 at 15:10
• Perfect answer! – Qwerty Oct 1 '16 at 15:17
• Could you write the definition of convergence in law your have at your disposal? The proof may be simpler or not according to the definition you have. – Davide Giraudo Oct 3 '16 at 9:44
• A sequence $X_1,X_2,.....$ of real-valued random variable is said to converge in distribution, or converge weakly, or converge in law to a random variable $X$ if $\lim_{n\to \infty} F_n(x)=F(x)$ for every number $x\in\Bbb{ R }$ at which F is continuous. – Qwerty Oct 3 '16 at 10:57

Yes $X_n$ converges to $X$ in law iff $E(h(X_n))$ converges to $E(h(X)) \forall h$ bounded and continuous from $\Bbb{R}$ to $\Bbb{R}$. Apply this on $g(X_n)$ and $g(X)$.