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.

edit: $X_n$ has density

$$p_n(x) = \frac{n-1}{n} g(x) + \frac{1}{n} g(x-n)$$

where $g$ is the density of the gaussian with unit variance and mean 0

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).

(note:the problem remains with probability measures)

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 ?

Original post here (crosspost crossvalidated!): https://stats.stackexchange.com/questions/186670/topologies-for-which-the-ensemble-of-probability-distributions-is-complete

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    $\begingroup$ The space of regular Borel probability measures on $\mathbb{R}$ is complete with respect to the total variation norm; by the Riesz representation theorem you can identify it with a closed subset of the dual of the Banach space $C_0(\mathbb{R})$, for instance. I'm a little confused by your example, but if the distributions really do converge in total variation then the limit is a regular Borel probability measure. $\endgroup$ – Paul Siegel Dec 17 '15 at 11:42
  • $\begingroup$ That's interesting. The sequence of $X_n$ do converge in TV (the distance is upper-bounded: $d(X_n,X_\infty)\leq 1/n$). Is being complete weirdly defined for measures ? $\endgroup$ – Guillaume Dehaene Dec 17 '15 at 15:41
  • $\begingroup$ Completeness isn't anything weird for measures, but the measures themselves can be weird. Could you elaborate on exactly what your $X_n$ are, perhaps by writing out their density functions? $\endgroup$ – Paul Siegel Dec 17 '15 at 15:51
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    $\begingroup$ What do you mean by what "the limit is really"? Under total variation, as you noted, the limit is really N(0,1). $\endgroup$ – Yoav Kallus Dec 20 '15 at 19:06
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    $\begingroup$ OK, I agree that your $X_n$ converge in TV to $N(0,1)$, and that the first two moments of $X_n$ converge to $1$ and $\infty$, respectively. But this is not an issue with completeness, it's an issue with continuity. I think you are really trying to ask: "Is there a topology on the space of probability distributions with the property that the functions $X \mapsto E(X)$ and $X \mapsto Var(X)$ defined in this space are continuous? $\endgroup$ – Paul Siegel Dec 20 '15 at 20:17

As I asserted in my comments, I think it is too much to hope for a reasonable topology on the space of random variables which makes the map $X \mapsto E(X)$ continuous. This is a bit like hoping that pointwise convergence or convergence in measure implies convergence in the $L^1$ norm; it seems reasonable, but there are simple counter-examples.

But all is not lost. In analysis one salvages the situation by making stronger assumptions: for instance, if $f_n \to f$ pointwise and $|f_n(x)| \leq g(x)$ for some integrable function $g$ and all $n$, then $\int f_n \to \int f$ (the dominated convergence theorem). There is a sort of counterpart to this in probability theory.

Definition: A sequence $X_n$ of random variables is uniformly integrable if:

  • $E(|X_n|)$ is uniformly bounded in $n$: there is a constant $K$ such that $E(|X_n|) \leq K$ for all $n$
  • For every $\varepsilon > 0$ there exists $\delta > 0$ such that $\int_A |X_n| < \varepsilon$ for all $n$ whenever $P(A) < \delta$

Uniform integrability is implied by the stronger (but more easily checked) condition that $E(|X_n|^{1 + \delta})$ is uniformly bounded for some $\delta > 0$.

Theorem: Suppose $X_n$ converges to $X$ in distribution and the sequence $|X_n|^k$ is uniformly integrable. Then $E(X_n^j) \to E(X^j)$ whenever $1 \leq j \leq k$. (Reference)

There are some other results of this flavor in the wikipedia page on convergence of random variables but this theorem is the best result that I know.

  • $\begingroup$ Uniform integrability reminds me a little of equicontinuity. I wonder if the theorem above is a special case of a suitable generalization of the Arzela-Ascoli theorem. $\endgroup$ – Paul Siegel Dec 20 '15 at 23:25
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    $\begingroup$ I sort of agree with the basic point, but the choice of example in your first paragraph is unfortunate, if I am not mistaken; the map $f\to \int f$ is continuous as a linear functional on the normed space $L^1({\bf R})$ $\endgroup$ – Yemon Choi Dec 21 '15 at 0:45
  • $\begingroup$ @YemonChoi Arg, continuity of that map is essentially a tautology. After recovering from flashbacks to my qualifying exams in graduate school, I amended the answer. $\endgroup$ – Paul Siegel Dec 21 '15 at 1:28
  • $\begingroup$ @PaulSiegel: thank you for the answer ! However, there is a simple metric topology which I think works (though, since I've forgotten my topology, I'm not completely sure what "making the map continuous" is exactly). That's the Wasserstein-1 metric. More generally, the Wasserstein-k metric (I think) works for all statistics with at most polynomial growth $\endgroup$ – Guillaume Dehaene Jan 4 '16 at 7:36

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