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From Wikipedia:

Let $\mu_n, n \in N$ be a sequence of probability measures on a metric space S; suppose that $\mu_n$ converges weakly to some probability measure $\mu$ on S as $n \to \infty$. Suppose also that the support of μ is separable. Then there exist random variables $X_n, X$ defined on a common probability space $(\Omega,F, P)$ such that

$X_n \xrightarrow{d}\ \mu_n$ (i.e. $μ_n$ is the distribution/law of $X_n$);

$X \xrightarrow{d}\ \mu$ (i.e. $\mu$ is the distribution/law of X); and

$X_n \xrightarrow{\mathrm{a.s.}} x$

Can someone provide a simple, concrete example of how one would use this theorem?

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One example is a version of the continuous mapping theorem which states that if $X_n \rightsquigarrow X$ then $f(X_n) \rightsquigarrow f(X)$ for a continuous function $f$. Using the a.s. representation (Skorohod's Representation theorem) there is a sequence of random variables $Y_n$ and a random variable $Y$ defined on a common probability space having the same laws as $X_n$ and $X$ s.t. $Y_n\xrightarrow{a.s.}Y$. The rest is pretty straightforward...

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  • $\begingroup$ Thanks, I guess I'd like to know why this theorem is so important. Its in all advanced probability textbooks, but I never really got its importance or why we need it. $\endgroup$ – user237392 Jun 28 '15 at 14:39
  • $\begingroup$ @Bey Which one? $\endgroup$ – d.k.o. Jun 28 '15 at 22:10
  • $\begingroup$ Skorohods theorem $\endgroup$ – user237392 Jun 28 '15 at 22:11
  • $\begingroup$ It's used to prove other results like any other theorem in math... $\endgroup$ – d.k.o. Jun 28 '15 at 22:13
  • $\begingroup$ Of course, but is it used to help calculate any probabilities or formulate models, or is it really useful for proving convergence properties? $\endgroup$ – user237392 Jun 29 '15 at 2:02

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