*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!

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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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