Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.