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Contents of PMS, Vol. 40, Fasc. 2,
pages 349 - 367
DOI: 10.37190/0208-4147.40.2.9
Published online 5.10.2020
 

On a relation between classical and free infinitely divisible transforms

Zbigniew J. Jurek

Abstract: We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-0ptselfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.

2010 AMS Mathematics Subject Classification: Primary 60E07, 60H05, 33B15; Secondary 44A10, 60B10.

Keywords and phrases: infinite divisibility, free-infinite divisibility, convolution semigroups, characteristic function, Voiculescu transform, Lévy–-Khinchin formula, Lévy (spectral) measure, Riemann zeta functions, Euler function, digamma function.

References


[1] A. Araujo E. Giné (1980), The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York.

[2] O. E. Barndorff-Nielsen S. Thorbjørnsen (2002), Self-decomposability and Lévy processes in free probability, Bernoulli 8, 323-366.

[3] H. Bercovici and V. Pata (1999), Stable laws and domains of attraction in free probability theory, Ann. of Math. 149, 1023-1060.

[4] H. Bercovici D. V. Voiculescu (1993), Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42, 733-773.

[5] P. Billingsley (1986), Probability and Measure, 2nd ed., Wiley, New York.

[6] I. S. Gradshteyn I. M. Ryzhik (1994), Table of Integrals, Series and Products, 5th ed., Academic Press, New York.

[7] T. Hasebe, N. Sakuma S. Thorbjørnsen (2019), The normal distribution is freely self-0ptdecomposable, Int. Math. Res. Notices, 1758-1787.

[8] J. Jacod (1985), Grossissements de filtration et processus d'Ornstein-Uhlenbeck generalisé, in: Grossissements de filtrations: exemples et applications, T. Jeulin and M. Yor (eds.), Lecture Notes in Math. 1118, Springer, 37–44.

[9] L. Jankowski Z. J. Jurek (2012), Remarks on restricted Nevanlinna transforms, Demonstratio Math. 45, 297-307.

[10] Z. J. Jurek (1996), Series of independent exponential random variables, in: Probability Theory and Mathematical Statistics (Tokyo, 1995), World Sci., 174-182.

[11] Z. J. Jurek (2006), Cauchy transforms of measures as some functionals of Fourier transforms, Probab. Math. Statist. 26, 187-200.

[12] Z. J. Jurek (2007), Random integral representations for free-infinitely divisible and tempered stable distributions, Statist. Probab. Lett. 77, 417-425.

[13] Z. J. Jurek (2007), On a method of introducing free-infinitely divisible probability measures, Demonstratio Math. 49, 236-251.

[14] Z. J. Jurek W. Vervaat (1983), An integral representation for self-decomposable Banach space valued random variables, Z. Wahrsch. Verw. Gebiete 62, 247-262.

[15] Z. J. Jurek M. Yor (2004), Selfdecomposable laws associated with hyperbolic functions, Probab. Math. Statist. 24, 181-191.

[16] K. R. Parthasarathy (1967), Probability Measures on Metric Spaces, Academic Press, New York.

[17] J. Pitman M. Yor (2003), Infinitely divisible laws associated with hyperbolic functions, Canad. J. Math. 55, 292-330.

[18] D. Voiculescu (1999), Lectures on free probability, in: Lectures on Probability Theory and Statistics (Saint-Flour, 1998), Springer, 279-349.

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