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Consider an i.i.d sequence $(X_i)_{i \geq 1}$ of scalar random variables distributed as $\pi$. It is not difficult to check that for almost every realization $(x_i)_{i \geq 1}$ of this sequence, the corresponding sequence of probability distributions $\pi_n := n^{-1} \sum_{i=1}^n \delta_{x_i}$ converges in distribution towards $\pi$. Any reference for this result?

Thanks!

[edit] as it turns out, this is more or less trivial in the scalar case, but it is easy to extend that to more general spaces. A possible proof consists in noting that (under mild assumption on the state space) almost surely the sequence $\pi_n$ is tight, and in this case it suffices to check (again, mild assumptions needed here) that $\pi_n(f) \to \pi(f)$ for every continuous function that has compact support, and by a separability argument one can conclude. Standard Glivenko-Cantelli type of arguments based on the cumulative distribution function seem slightly awkward, but I suspect that this is a classical result anyway. References?

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Glivenko-Cantelli theorem. There are also generalisations to non-scalar settings, google "empirical processes".

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  • $\begingroup$ great, that's perfect. Thank you. $\endgroup$
    – Alekk
    Commented Feb 13, 2014 at 1:35

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