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

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.