The Mittag-Leffler function in the thinning theory for renewal processes
Authors:
R. Gorenflo and F. Mainardi
Journal:
Theor. Probability and Math. Statist. 98 (2019), 105-113
MSC (2010):
Primary 33E12, 60K05, 60K25; Secondary 26A33, 76R50
DOI:
https://doi.org/10.1090/tpms/1065
Published electronically:
August 19, 2019
MathSciNet review:
3824681
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Abstract: The main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law-type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in the theory of continuous time random walk and consequently of fractional diffusion in a plenary lecture by the late Professor Gorenflo at a Seminar on Anomalous Transport held in Bad-Honnef in July 2006, published in a 2008 book. After recalling the basic theory of renewal processes including the standard and the fractional Poisson processes, here we have revised the original approach by Gnedenko and Kovalenko for the convenience of the experts of stochastic processes who are not aware of the relevance of the Mittag-Leffler functions.
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References
- L. Beghin and E. Orsingher, Iterated elastic Brownian motions and fractional diffusion equations, Stochastic Process. Appl. 119 (2009), 1975–2003. MR 2519353
- F. Beichelt, Stochastic Processes in Science, Engineering and Finance, Chapman & Hall/CRC, Boca Raton, US, 2006. MR 2207217
- M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent, Part II, Geophys. J. R. Astr. Soc. 13 (1967), 529–539.
- M. Caputo, Elasticità e Dissipazione, Bologna, Zanichelli, 1969.
- M. Caputo and F. Mainardi, Linear models of in anelastic solids, Riv. Nuovo Cimento (Ser. II) 1 (1971), 161–198.
- D. R. Cox, Renewal Theory, 2nd ed., Methuen, London, 1967. MR 0153061
- L. Devroye, A note on Linnik’s distribution, Statist. Probab. Lett. 9 (1990), 305–306. MR 1047827
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Bateman Project, McGraw-Hill, New York, 1955, vol. 3, pp. 206–227.
- W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, 2nd ed., Wiley, New York, 1971. MR 1570945
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- F. Mainardi, R. Gorenflo, and E. Scalas, A fractional generalization of the Poisson processes, Vietnam J. Math. 32 SI (2004), 53–64. MR 2120631
- K. V. Mitov and E. Omey, Renewal Processes, Springer, Heidelberg, 2014. MR 3308384
- R. N. Pillai, On Mittag-Leffler functions and related distributions, Ann. Inst. Statist. Math. 42 (1990), no. 1, 157–161. MR 1054728
- I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. MR 1658022
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- T. Szàntai, On an invariance problem related to different rarefactions of recurrent event, Studia Sci. Math. Hungar. 6 (1971), 453–456. MR 0293736
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Additional Information
R. Gorenflo
Affiliation:
Department of Mathematics and Informatics, Freie Universität, Berlin, Germany
F. Mainardi
Affiliation:
Department of Physics and Astronomy, University of Bologna, Italy
Email:
francesco.mainardi@bo.infn.it (Corresponding Author)
Keywords:
Mittag-Leffler functions,
thinning (rarefaction),
renewal processes,
queuing theory,
Poisson process
Received by editor(s):
March 14, 2018
Published electronically:
August 19, 2019
Additional Notes:
The work of the second author was carried out in the framework of the activities of the National Group of Mathematical Physics (GNFM, INdAM)
Dedicated:
This note is dedicated to the memory of the late Professor Rudolf Gorenflo, who passed away on 20 October 2017 at the age of 87.
Article copyright:
© Copyright 2019
American Mathematical Society
