Course on

The eleventh and twelveth problems of Rota's Fubini Lectures: from cumulants to free probability theory

by

Domenico Senato & Elvira di Nardo

(Università della Basilicata, Italia)

22 - 24 March 2010

The lectures will focus on the following topic: the development of the classical umbral calculus, as featured by Rota and Taylor in 1994. Thanks to this symbolic method, an unifying theory of classical, boolean and free cumulants has been carried out. This unifying theory relies on a suitable representation of Abel polynomials via classical umbral calculus.

Through this representation, we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. We give a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for all kind of cumulants and it is particularly suited to be implemented by using software for symbolic computations, as for example MAPLE. Comparisons with existing procedures, especially for conversions between moments and free cumulants, show the efficiency of the proposed approach.

Some in-between steps lead us to develop some topics of computational statistics, which benefit of this symbolic approach in efficiency and speeding up. We resume also these applications.

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