Saturday, August 11, 2012

The Law of Large Numbers for independent identically distributed nonmeasurable random variables

Fact: For any real-valued function f, measurable or not, on a probability space, there exists a largest measurable function fL such that fLf and a smallest measurable function fU such that fUf, and fL and fU are unique up to almost sure equality.

Definition: A set U in a probability space is maximally nonmeasurable providing all its measurable subsets have measure zero and all its measurable supersets have measure one.

Definition: A sequence X1,X2,... of independent identically distributed not necessarily measurable random variables will be a sequence of functions on an infinite product of copies of a probability space, such that Xn(w1,w2,...)=F(wn) for each n and a single fixed function F.

Henceforth suppose X1,X2,... are like that. Let Sn=X1+...+Xn.

Easy consequence of the Law of Large Numbers: If X1L and X1U have finite expectations, then almost surely E[X1L]≤ liminf Sn/n≤ limsup Sn/nE[X1U].

Can one strengthen this? E.g., can one hope that one of the inequalities is an equality? Yesterday I finished proving a negative answer.

Theorem: Suppose X1L and X1U are integrable. Let A be any proper non-empty subset of the interval [E[X1L],E[X1U]] (which implies that E[X1L]<E[X1U]). Consider the respective subsets of our probability space where:

  • lim Sn/n exists
  • lim Sn/n exists and is in A
  • limsup Sn/n is in A
  • liminf Sn/n is in A
  • all the limit points of Sn/n are in A
Then each of these subsets is maximally nonmeasurable.

This has a very interesting consequence for the philosophy of science, namely that unless we assume at the outset that what we are observing in the real world are measurable random variables, we can never come to that conclusion on the basis of observation of frequencies. For non-trivial cases (i.e., ones where E[X1L]<E[X1U]) of nonmeasurable random variables can equally well give neat limiting frequencies and not give them—any such limiting outcome is itself probabilistically maximally nonmeasurable.


WMF said...

This sounds intriguing. I'll have to think about this. Just for reference, is the first fact by any chance known by another name?

Alexander R Pruss said...

I wouldn't be surprised if there is a proof of the first fact in something by Hoffman-Jorgensen. But I didn't find a reference, so I just proved it in the paper (temporary link).

Alexander R Pruss said...

I found a reference for the fact (and other related facts). See Lemmas 1.2.2 and 1.2.3 in A. van der Vaart and J. Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics, New York: Springer, 1996

Alexander R Pruss said...

Here is a permanent link to the paper.

Alexander R Pruss said...

This paper has just been accepted by the Bulletin of the Polish Academy of Sciences, Mathematics.