## Monday, 21 February 2011

### Algebraic, metric and probabilistic properties of convex combinations based on the t-normed extension principle: the Strong Law of Large Numbers

P. Terán (2013). Fuzzy Sets and Systems 223, 1-25.

I think this is a nice paper. It combines many things and, being a 42-page manuscript to prove one theorem, it brings an `epic' culmination to part of my earlier research.

The general framework is that of convex combination spaces, an attempt by Ilya Molchanov and myself (2006 below) at an axiomatic treatment of expectation in metric spaces taking the convex combination of points as the basic operation.

Consider fuzzy sets in such a metric space. The convex combination operation in the carrier space can be uplifted to the superspace of fuzzy sets by using one of many extension principles in the literature. These extension devices differ in the choice of a continuous triangular norm (a special ordered topological semigroup in [0,1] ).

I had already showed that, with an appropriate topology, any such extension device satisfies the Strong Law of Large Numbers (in a finite-dimensional space; 2008 below). The fact that the limit in the SLLN varies with the triangular norm was the motivation to go for the abstract result of convex combination spaces.

The problem is for which triangular norms the SLLN holds in another topology which is the strongest in the literature. It was known that the minimum yields such an SLLN, and the product does not.

It turns out that the stronger SLLN is characterized
-algebraically: SLLN iff the triangular norm is eventually idempotent.
-metrically: SLLN iff the extension to fuzzy sets retains the property of being a convex combination space.

A nice result!

Up the line:
·On limit theorems for t-normed sums of fuzzy random variables (2004).
·A law of large numbers in a metric space with a convex combination operation (2006, w. Ilya Molchanov). Downloadable from Ilya's website.
·Probabilistic foundations for measurement modelling with fuzzy random variables (2007).

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

Down the line:
There remains a short coda.