## Continuous Mapping Theorem for sequences of random variables

Hi everyone,

I need help proving the CMT for random vectors. I'm currently reading Econometric Analysis for Cross Section and Panel Data by Jeffrey M. Wooldridge, and, unfortunately, he leaves it to the reader to prove most of the asymptotic results.

Q. Prove that if x_n converges in distribution to x, then for a continuous function g: R^k to R^l, g(x_n) converges in distribution to g(x). I need a rigorous, epsilon-delta proof of this. Any good, rigorous links to a proof would be appreciated.

Thanks. Christian S.

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 Why do you need a rigorous $\epsilon$-$\delta$ proof? – David Roberts Oct 29 at 6:45 Well, I want to know how exactly they derive this result. Almost every textbook I have simply states the result. I've tried doing this myself, writing out the exact definition of distributional convergence in terms of epsilons and deltas, and then try to use properties of continuous functions, inverse images, etc., to get the result out. Hasn't worked. By the way, I am a economics/econometrics student, so my understanding of measure-theoretic statistics is limited. But I would like someone to try their best to prove this, or to direct me to a proof which spells everything out. Christian – – Christian Silvestro Oct 29 at 7:49 This question does not seems fit for MO; I guess you should ask it at Math.SE. – Benoît Kloeckner Oct 29 at 12:20