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?

  • 4
    $\begingroup$ Yes; this is known as the continuous mapping or Mann-Wald theorem. $\endgroup$ – Math1000 Oct 1 '16 at 15:10
  • $\begingroup$ Perfect answer! $\endgroup$ – Qwerty Oct 1 '16 at 15:17
  • $\begingroup$ 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. $\endgroup$ – Davide Giraudo Oct 3 '16 at 9:44
  • $\begingroup$ 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. $\endgroup$ – 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 $g(X)$.

  • $\begingroup$ Aar ektu explain korle bhalo hoto. $\endgroup$ – Qwerty Oct 16 '16 at 18:33
  • $\begingroup$ Aar PR ki convergence in law'r ei definition ta bolechilo? $\endgroup$ – Qwerty Oct 16 '16 at 18:38
  • $\begingroup$ Ha bolechilo... Btw tui ke 😅 $\endgroup$ – user379195 Oct 16 '16 at 19:08
  • $\begingroup$ Ekhane bolbo na. Kalke bole debo. $\endgroup$ – Qwerty Oct 16 '16 at 19:11

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