Perleseminar: Professor Steen Thorbjørnsen, Aarhus University
27.09.2019 kl. 12.15 - 13.15
"Unimodality of the freely selfdecomposable probability laws"
A finite measure m on the real line is called unimodal, if, loosely speaking, it has a Lebesgue density which in increasing on (-oo, a] and decreasing on [a,oo) for some real number a. In addition m may have a single atom at a. In 1978 M. Yamazato settled in the positive the long standing conjecture on unimodality of the selfdecomposable probability laws. In doing so Yamasato also gave the first full proof of the unimodality of the stable distributions.
In the early 1980's Voiculescu introduced the notion of freeness, which is an alternative notion of independence for (quantum) random variables. Since then a full fledged probability theory, parallel to the classical one, has been developed based on the notion of freeness. In 1999 P. Biane proved that the stable distributions in free probability are unimodal, and in recent joint work with T. Hasebe the speaker established that the same conclusion in fact holds for all selfdecomposable laws in free probability. The talk will present an outline of the proof of the latter result and describe some of the involved techniques based on Stieltjes inversion.
Institut for Matematiske Fag, Aalborg Universitet
Skjernvej 4A, rum 5.034