A connection between free and classical infinite divisibility

TY - JOUR

T1 - A connection between free and classical infinite divisibility

AU - Barndorff-Nielsen, Ole Eiler

AU - Thorbjørnsen, Steen

PY - 2004

Y1 - 2004

N2 - In this paper we continue our studies, initiated in Refs. 2–4, of the connections between the classes of infinitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely infinitely divisible probability measure equals the classical cumulant transform of a certain classically infinitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We find, furthermore, an alternative definition of the Bercovici–Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Lévy–Khintchine representations (classical and free, respectively).

AB - In this paper we continue our studies, initiated in Refs. 2–4, of the connections between the classes of infinitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely infinitely divisible probability measure equals the classical cumulant transform of a certain classically infinitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We find, furthermore, an alternative definition of the Bercovici–Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Lévy–Khintchine representations (classical and free, respectively).

U2 - 10.1142/S0219025704001773

DO - 10.1142/S0219025704001773

M3 - Journal article

VL - 7

SP - 573

EP - 590

JO - Inf. Dim. Anal. Quantum Prob. Rel. Topics

JF - Inf. Dim. Anal. Quantum Prob. Rel. Topics

IS - 4

ER -