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I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess.

For example, consider a mixture random variable $X_n$: pick a Gaussian centered at 0 with variance 1, and with probability $\frac{1}{n}$, add $n$ to the result. A sequence of such random variables would converge (weakly and in total variation) to a Gaussian centered at 0 with variance 1, but the mean of the $X_n$ is always $1$ and the variances converge to $+\infty$. I really don't like saying that this sequence converges because of that.

I took me quite some time to remember everything I've forgotten about topologies, but I finally figured out what was so unsatisfying to me about such examples: the limit of the sequence is not a conventional distribution. In the example above, the limit is a weird "Gaussian of mean 1 and of infinite variance". In topological terms, the set of probability distributions isn't complete under the weak (and TV, and all the other topologies I've looked at).

I then face the following question:

  • does there exist a topology such that the ensemble of probability distributions is complete ?

  • If no, does that absence reflect an interesting property of the ensemble of probability distributions ? Or is it just boring ?

Note: I have phrased my question about "probability distributions". These can't be closed because they can converge to Diracs and stuff like that which don't have a pdf. But measures still aren't closed under the weak topology so my question remains

crossposted to mathoverflow https://mathoverflow.net/questions/226339/topologies-for-which-the-ensemble-of-probability-measures-is-complete?noredirect=1#comment558738_226339

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    $\begingroup$ You discovered that the set of all probability distributions is noty compact. I think compactness is the word you need, not completeness. The relevant concept of compactness in this setting is often called tightness. See for instance stats.stackexchange.com/questions/180139/… $\endgroup$ – kjetil b halvorsen Dec 17 '15 at 16:03
  • $\begingroup$ @kjetilbhalvorsen I think it is precompact instead of compact due to Skorohod's Theorem. $\endgroup$ – Henry.L Mar 15 '17 at 19:09
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Looking at the question from a more narrow statistical angle (the general mathematical topological issue is valid), the fact that the sequence of moments may not converge to the moments of the limiting distribution is a well-known phenomenon. This in principle, does not automatically set in doubt the existence of a well behaved limiting distribution of the sequence.

The limiting distribution of the above sequence $\{X_n + n Bern(1/n)\}$ is a well-behaved $N(0,1)$ distribution with finite moments. It is the sequence of the moments that does not converge. But this is a different sequence, a sequence comprised of functions of our random variables (integrals, densities and such), not the sequence of the random variables themselves whose limiting distribution we are interested at.

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  • $\begingroup$ How does this answer the question? $\endgroup$ – whuber Nov 20 '17 at 18:46
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    $\begingroup$ @whuber Well, my answer says that whether there exists such a topology as the OP asks for, or not, makes not much difference from a statistical point of view. $\endgroup$ – Alecos Papadopoulos Nov 20 '17 at 19:07

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