Mandelbrot, fractals and counterexamples in applied probability

Benoît Mandelbrot died yesterday. Like most of the blogs dealing with applied mathematics, it looks like I have to mention this event. Unfortunately, I don’t know much about fractals…

The first time I heard about Mandelbrot and chaos was when I have been working on fractional time series (see eghere). Murad Taqqu gave a very interesting short course in Paris, and I have been using it in two papers (actually one more should appear soon in Climate Change).

The second time was in Québec (city), five years ago, when Roger Nelsen gave a talk on copulas with fractal suppport (here). By that time, we were finishing our paper with Alessandro (in mathematical finance, and limiting theorems). I remember adding a Remark in the paper (that can be found here), since using that kind of copulas was a nice way to show that, without sufficient regularity conditions, the limit we were looking for had no sense.

A few months after, with Johan, in a paper on pitfalls on lower tail dependence for Archimedean copulas (here), we used again this fractal construction (here applied to Archimedean copulas) to find a nice counterexample to a (false) theorem on regular variation. Again, the goal was to understand how the dependence structure of https://perso.univ-rennes1.fr/arthur.charpentier/latex/mandel01.png given https://perso.univ-rennes1.fr/arthur.charpentier/latex/mandel02.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/mandel03.png changed when https://perso.univ-rennes1.fr/arthur.charpentier/latex/mandel04.png goes to 0. In the case of Archimedean copulas, the copula of the conditional pair is still Archimedean (with another generator, except for Clayton copula). The graph below show how https://perso.univ-rennes1.fr/arthur.charpentier/latex/mandel05.png changes, as https://perso.univ-rennes1.fr/arthur.charpentier/latex/mandel04.png decreases… Actually, I draw https://perso.univ-rennes1.fr/arthur.charpentier/latex/mandel06.png since Archimedean generators are not unique.

I have also decided to plot https://perso.univ-rennes1.fr/arthur.charpentier/latex/mandel07.png.

Here, we see that there is no way of talk about a possible limit for the conditional copula because of a fractal behavior in 0 of the generator (even if my fractal are not as nice as the one you can find on the internet….). So thanks Benoît for giving us a nice toy to build interesting counterexamples !


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