# Topologies for which the ensemble of probability distributions is complete

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

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.