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A Lifetime of Excursions Through Random Walks and Lévy Processes Read chapter Read first chapter

2021 | OriginalPaper | Chapter

On Doney’s Striking Factorization of the Arc-Sine Law

Authors : Larbi Alili, Carine Bartholmé, Loïc Chaumont, Pierre Patie, Mladen Savov, Stavros Vakeroudis

Published in: A Lifetime of Excursions Through Random Walks and Lévy Processes

Publisher: Springer International Publishing

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Abstract

In Doney (Bull Lond Math Soc 19(2):177–182, 1987), R. Doney identifies a striking factorization of the arc-sine law in terms of the suprema of two independent stable processes of the same index by an elegant random walks approximation. In this paper, we provide an alternative proof and a generalization of this factorization based on the theory recently developed for the exponential functional of Lévy processes. As a by-product, we provide some interesting distributional properties for these variables and also some new examples of the factorization of the arc-sine law.

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previous chapter Path Decompositions of Perturbed Reflecting Brownian Motions
next chapter On a Two-Parameter Yule-Simon Distribution
Literature
1.
Alili, L., Jedidi, W., Rivero, V.: On exponential functionals, harmonic potential measures and undershoots of subordinators. ALEA Lat. Am. J. Probab. Math. Stat. 11(1), 711–735 (2014)MathSciNetMATH
2.
Bartholmé, C.: Self-similarity and exponential functionals of Lévy processes. PhD thesis, Université Libre de Bruxelles (2014)
3.
4.
Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)
5.
6.
Bondesson, L.: On univariate and bivariate generalized gamma convolutions. J. Stat. Plan. Inference 139(11), 3759–3765 (2009)MathSciNetCrossRef
7.
Caballero, M.E., Chaumont, L.: Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43(4), 967–983 (2006)MathSciNetCrossRef
8.
Chaumont, L., Yor, M.: Exercises in Probability. A Guided Tour from Measure Theory to Random Processes, Via Conditioning, 2nd edn. Cambridge University Press, Cambridge/New York (2012)
9.
Doney, R.A.: On the maxima of random walks and stable processes and the arc-sine law. Bull. Lond. Math. Soc. 19(2), 177–182 (1987)MathSciNetCrossRef
10.
Doney, R.A.: Fluctuation Theory for Lévy Processes. Ecole d’Eté de Probabilités de Saint-Flour XXXV-2005. Springer, Berlin (2007)
11.
Doney, R.A., Savov, M.: The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab. 38(1), 316–326 (2010)MathSciNetCrossRef
12.
Jacob, N.: Pseudo Differential Operators and Markov Processes. Vol. I, Fourier Analysis and Semigroups. Imperial College Press, London (2001)CrossRef
13.
Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. A Wiley-Interscience Publication, vol. 2, 2nd edn. Wiley, New York (1995)
14.
15.
Kuznetsov, A., Pardo, J.C.: Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123, 113–139 (2013)MathSciNetCrossRef
16.
Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin (2006) KSK
17.
Kyprianou, A.E.: Stable Lévy processes, self-similarity and the unit ball. ALEA Lat. Am. J. Probab. Math. Stat. 15(1), 617–690 (2018)MathSciNetCrossRef
18.
Lebedev, N.N.: Special Functions and Their Applications. Dover Publications Inc., New York (1972). Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication
19.
20.
Markushevich, A.I.: Entire functions. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. American Elsevier Publishing Co., Inc., New York (1966)
21.
22.
Pitman, J., Yor, M.: Arcsine laws and interval partitions derived from a stable subordinator. Proc. Lond. Math. Soc. 65(3), 326–356 (1992)MathSciNetCrossRef
23.
24.
25.
Patie, P.: Law of the absorption time of some positive self-similar Markov processes. Ann. Probab. 40(2), 765–787 (2012)MathSciNetCrossRef
26.
Patie, P., Savov, M.: Extended factorizations of exponential functionals of Lévy processes. Electron. J. Probab. 17(38), 22 (2012)MATH
27.
Patie, P., Savov, M.: Bernstein-gamma functions and exponential functionals of Lévy processes. Electron. J. Probab. 23(75), 1–101 (2018)MATH
28.
Patie, P., Savov, M.: Spectral expansion of non-self-adjoint generalized Laguerre semigroups. Mem. Am. Math. Soc. to appear, 1–179 (2019)
29.
Sato, K.-I.: Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999). Translated from the 1990 Japanese original, Revised by the author
30.
Spitzer, F.: A combinatorial lemma and its application to probability theory. Trans. Am. Math. Soc. 82, 323–339 (1956)MathSciNetCrossRef
31.
Vakeroudis, S., Yor, M.: A scaling proof for Walsh’s Brownian motion extended arc-sine law. Electron. Commun. Probab. 17(63), 1–9 (2012)MathSciNetMATH
32.
Yor, M.: Exponential functionals of Brownian motion and related processes. Springer Finance, Berlin (2001)CrossRef
Metadata
Title
On Doney’s Striking Factorization of the Arc-Sine Law
Authors
Larbi Alili
Carine Bartholmé
Loïc Chaumont
Pierre Patie
Mladen Savov
Stavros Vakeroudis
Copyright Year
2021
Book
A Lifetime of Excursions Through Random Walks and Lévy Processes

Print ISBN: 978-3-030-83308-4

Electronic ISBN: 978-3-030-83309-1

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-83309-1_3