Thursday, 24 October 2013

A law of large numbers for the possibilistic mean value

P. Terán (2014). Fuzzy Sets and Systems 245, 116-124.

This paper has an interesting idea, I think.

In general, random variables in a possibility (instead of probability) space do not satisfy a law of large numbers exactly like the one in probability theory. The reason is that, if the variable takes on at least two different values x, y with full possibility, it is fully possible that the sample average equals x for every sample size and also that it equals y. Thus we cannot ensure that the sample average converges to either x or y necessarily.

The most you can say, with Fullér and subsequent researchers, is that the sample average will necessarily tend to being confined within a set of possible limits.

The interesting idea is the following:
1) Define a suitable notion of convergence in distribution.
2) Show that the law of large numbers does hold in the sense that the sample averages converge to a random variable in distribution, even if it cannot converge in general either in necessity or almost surely.
3) Show that, magically, the statement in distribution is stronger than the previous results in the literature, not weaker as one would expect!

On a more anecdotal note, this paper was written as a self-challenge. Physicist Edward Witten is said to type (or have typed some) papers by improvisation, making the details up as he goes along. I have always considered myself uncapable of doing something like that, but I've been happy to learn that I was wrong.

Up the line:
·Strong law of large numbers for t-normed arithmetics (2008)
·On convergence in necessity and its laws of large numbers (2008)

Down the line:
There is a couple of papers under submission, and a whole lot of ideas.

To download the paper, click on the title or here

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