# Recovery of probability distribution from a single point

There is probably terminology for this, and I apologize that I don't know it, and part of my question is what the standard terminology for the concepts I'm giving is. This is a pretty open-ended question.

Let $C$ be some class of random variables, on the the same or different probability spaces, taking values in some topological space $V$ (the case that interests me most is $R^\omega$).

Let $D_C$ be the set of distributions of the variables in $C$, i.e., equivalence classes under probability isomorphism ($X:\Omega\to V$ and $Y:\Upsilon\to V$ are equivalent iff there is a probability isomorphism $\alpha:\Omega\to\Upsilon$ such that $X = Y\circ \alpha$).

Say that $C$ is "single-point recoverable" provided that there is a function $f:V\to D_C$ such that for every $X\in C$, $f(X)=[X]$ with probability one. I.e., for all $X\in C$, if $X:\Omega\to V$, then $f(X(\omega)) = [X]$ for almost all $\omega\in\Omega$.

Question 1: Is there a standard term for "single-point recoverable"?

Fact: If $V=\mathbb R^\infty$ (with product topology and a Borel $\sigma$-algebra) and $C_n$ is the class of all vector-valued random variables $X$ of the form $X=(X_i)_{i\in \mathbb N}$ where the $X_i$ are i.i.d. and have values in $R^n$, then $C_n$ is single-point recoverable. (Proof sketch: Rectangles are a Vapnis-Chervonenkis class, and we can a.s. recover the distribution of $X_1$, and hence of $X$, via a uniform strong LLN by using the frequencies of visits of $X_i(\omega)$ to rectangles.)

Question 2: Is there a standard name for the Fact? An elementary proof? (I may be missing something completely obvious.)

(The Fact is a sort of partial vindication of frequentism: It shows that if we know that an infinite sequence of observations was generated by an i.i.d. sequence, a.s. we can recover the distribution from the observations.)

Question 3: Are there any interesting generalizations of the Fact? For instance, if $E$ is the class of all sequences of $\mathbb R^n$-valued i.r.v.'s, then $E$ isn't single-point recoverable. But is there any result like this: There is a function $f:\mathbb R^\infty \to D_C$ such that for all $X\in E$, for almost all $\omega$, $f(X(\omega))$ is "asymptotically close" to $[X]$, in some precise sense? (Or at least for some subclass of $E$.) Is there any result that weakens independence?

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Regarding Question 1, I think the notion you are looking for is that $C$ consists of random variables having mutually singular distribution measures.