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From Combinatorics to Philosophy pp 91–130Cite as

The Eleventh and Twelveth Problems of Rota’s Fubini Lectures: from Cumulants to Free Probability Theory

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Abstract

The title of the paper refers to the Fubini Lecture “Twelve problems in probability no one likes to bring up”, delivered by Gian-Carlo Rota at the Institute for Scientific Interchange (Turin - Italy) in 1998. The eleventh and the twelfth problems of that lecture deal with cumulants and free probability theory respectively. Our aim is to describe the development of the classical umbral calculus, as featured by Rota and Taylor in 1994, thanks to which a unifying theory of classical, Boolean and free cumulants has been carried out. This unifying theory relies on an updating of Sheffer sequences via the classical umbral calculus. Some in-between steps lead us to develop some topics of computational statistics, which benefit of this symbolic approach in efficiency and speed-up. We resume also these applications.

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  1. Università degli Studi della Basilicata, Basilicata, Italy

    Elvira Di Nardo & Domenico Senato

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  1. Elvira Di Nardo
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  2. Domenico Senato
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Correspondence to Elvira Di Nardo .

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Editors and Affiliations

  1. Dipto Tecnologie Informatiche, Università Milano, Via Bramante, 65, 26013, Crema, Italy

    Ernesto Damiani

  2. Dipto. Informatica e Comunicazione, Università Milano, Via Comelico, 39, 20135, Milano, Italy

    Ottavio D’Antona  & Vincenzo Marra  & 

  3. Dipto. Filosofia, Università della Calabria, Ponte Bucci Cubo 18/c, 87036 Arcavacata di Rende (Cs), Cosenza, Italy

    Fabrizio Palombi

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Nardo, E.D., Senato, D. (2009). The Eleventh and Twelveth Problems of Rota’s Fubini Lectures: from Cumulants to Free Probability Theory. In: Damiani, E., D’Antona, O., Marra, V., Palombi, F. (eds) From Combinatorics to Philosophy. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-88753-1_6

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