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REGENERATIVE MUTATION PROCESSES RELATED TO THE SELFDECOMPOSABILITY OF SIBUYA DISTRIBUTIONS

Published online by Cambridge University Press:  22 June 2018

Thierry Huillet  and
Servet Martínez
Thierry Huillet
Affiliation:
Laboratoire de Physique Théorique et Modélisation CNRS UMR-8089, Site de Saint Martin, 2 avenue Adolphe-Chauvin, Université de Cergy-Pontoise Cergy-Pontoise 95302, France E-mail: Thierry.Huillet@u-cergy.fr
Servet Martínez
Affiliation:
Depto. Ingenieria Matematica and Centro Modelamiento Matematico Universidad de Chile UMI 2071, Uchile-Cnrs, Santiago, Casilla 170-3 Correo 3, Chile
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Abstract

The Sibuya distribution is a discrete probability distribution on the positive integers which, while Poisson-compounding it, gives rise to the discrete-stable distribution of Steutel and van Harn. We first address the question of the discrete self-decomposability of Sibuya and Sibuya-related distributions. Discrete self-decomposable distributions arise as limit laws of pure-death branching processes with immigration, translating a balance between immigration events and systematic ageing and ultimate death of the immigrants at constant rate. Exploiting this fact, we design a new Luria–Delbrück-like model as an intertwining of a coexisting two-types (sensitive and mutant) population. In this model, a population of sensitive gently grows linearly with time. Mutants appear randomly at a rate proportional to the sensitive population size, very many at a time and with Sibuya-related distribution; each mutant is then immediately subject to random ageing and death upon appearance. The zero-set of the times free of mutants, when the sensitive population lacks immunity, is investigated using renewal theory. Finally, assuming each immigrant to die according to a critical binary branching processes, now with heavy-tailed extinction times, we observe that the local extinction events can become sparse, leading to a congestion of the mutants in the system.

Keywords

discrete self-decomposabilityheavy-tailsmutation model of viral resistanceregenerative processesSibuya-related distributions
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 2 , April 2019 , pp. 291 - 325
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Arratia, R., Liggett, T.M. & Williamson, M.J. (2014). Scale-free and power law distribution via fixed points and convergence of (thinning and conditioning) transformations. Electronic Communications in Probability 19(39): 10.Google Scholar
2.Bondesson, L. (1992). Generalized Gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics, 76. New York: Springer.Google Scholar
3.Bondesson, L. (2015). A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables. Journal of Theoretical Probability 28: 1063–1081.Google Scholar
4.Bouzar, N. (2008). Semi-self-decomposable distributions on ℤ+. Annals of the Institute of Statistical Mathematics. 60(4): 901–917.Google Scholar
5.Bouzar, N. & Satheesh, S. (2008). Comments on α-decomposability. Metron - International Journal of Statistics LXVI(2): 243–252.Google Scholar
6.Christoph, G. & Schreiber, K. (1998). The generalized discrete Linnik distributions. In Kahle, W., Collani, E.v., Franz, J. & Jensen, U. (eds.), Advances in stochastic models for reliability, quality and safety. Statistics for Industry and Technology. Boston: Birkhäuser, pp. 3–18.Google Scholar
7.Christoph, G. & Schreiber, K. (1998). Discrete stable random variables. Statistics & Probability Letters 37: 243–247.Google Scholar
8.Christoph, G. & Schreiber, K. (2000). Scaled Sibuya distribution and discrete self-decomposability. Statistics & Probability Letters 48: 181–187.Google Scholar
9.Christoph, G. & Schreiber, K. (2001). Positive Linnik and discrete Linnik distributions. In Balakrishnan, N., Ibragimov, I.A. & Nevzorov, V.B. (eds.), Asymptotic methods in probability and statistics with applications. Statistics for Industry and Technology. Boston, MA: Birkhäuser, pp. 3–18.Google Scholar
10.Comtet, L. (1970). Analyse combinatoire Tomes 1 et 2. Paris: Presses Universitaires de France.Google Scholar
11.Dacey, M.F. (1969). A Hypergeometric Family of Discrete Probability Distributions: Properties and Applications to Location Models. Geographical Analysis 1(3): 219–320.Google Scholar
12.Devroye, L. (1990). A note on Linnik distribution. Statistics & Probability Letters 9: 305–306.Google Scholar
13.Devroye, L. (1993). A triptych of discrete distributions related to stable law. Statistics & Probability Letters 18: 349–351.Google Scholar
14.Feller, W. (1971). An introduction to probability theory and its applications. Vol.2, 2nd ed. New-York, London, Sydney: John Wiley & Sons.Google Scholar
15.Greenwood, M. (1931). On the statistical measure of infectiousness. Journal of Hygiene 31: Cambridge, 336–351.Google Scholar
16.Harris, T.E. (1963). The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119, Berlin; Englewood Cliffs, N.J: Springer-Verlag, Prentice-Hall, Inc.Google Scholar
17.Huillet, T. (2000). On Linnik's continuous-time random walks. Journal of Physics A: Mathematical & General 33: 26–31.Google Scholar
18.Karlin, S. & Taylor, H.M. (1975). A first course in Stochastic Process. 2nd ed. New York, London: Academic Press.Google Scholar
19.Keller, P. & Antal, T. (2015). Mutant number distribution in an exponentially growing population. Journal of Statistical Mechanics: Theory and Experiment 2015: P01011.Google Scholar
20.Kemp, C.D. & Kemp, A.W. (1956). Generalized Hypergeometric Distributions. Journal of the Royal Statistical Society. Series B. 18(2): 202–211.Google Scholar
21.Kozubowski, T.J. & Podgórski, K. (2017) A generalized Sibuya distribution. Annals of the Institute of Statistical Mathematics, online first. DOI 10.1007/s10463-017-0611-3.Google Scholar
22.Linnik, Y.V. (1953). Linear forms and statistical criteria. I, II, Selected Translations in Mathematical Statistics and Probability 3: 1–90, 1963, translated from Ukrainian Mathematical Journal 5, 207–243 and 247–290.Google Scholar
23.Lea, D.E. & Coulson, C.A. (1949). The distribution of the numbers of mutants in bacterial populations. Journal of Genetics 49(3): 264–285.Google Scholar
24.Luria, S.E. & Delbrück, M. (1943). Mutations of bacteria: from virus sensitivity to virus resistance. Genetics 28(6): 491–511.Google Scholar
25.Parzen, E. (1962). Stochastic processes. San Fransisco: Holden-Day, Inc., pp. 324.Google Scholar
26.Pillai, R.N. (1990). On Mittag-Leffler-functions and related distributions. Annals of the Institute of Statistical Mathematics 42: 157–161.Google Scholar
27.Pillai, R.N. & Jayajumar, K. (1995). Discrete Mittag-Leffler distributions. Statistics & Probability Letters 23(3): 271–274.Google Scholar
28.Pitman, J. (2006) Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 724, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Berlin: Springer-Verlag.Google Scholar
29.Quine, M.P. & Seneta, E. (1969). A limit theorem for the Galton-Watson process with immigration. Australian & New-Zealand Journal of Statistics 11(3): 166–173.Google Scholar
30.Ross, S.M. (1983). Stochastic Processes. New York: John Wiley and Sons, Inc.Google Scholar
31.Ross, S.M. (2010). Introduction to Probability Models. 10th ed. Amsterdam: Elsevier/Academic Press.Google Scholar
32.Sapatinas, T. (1995). Characterizations of probability distributions based on discrete p−monotonicity. Statistics & Probability Letters 24: 339–344.Google Scholar
33.Schreiber, K. (1999) Discrete self-decomposable distributions. Dr. rer. nat. Thesis dissertation. Otto-von-Guericke-Universität Magdeburg.Google Scholar
34.Sibuya, M. (1979) Generalized hypergeometric, digamma and trigamma distributions. Annals of the Institute of Statistical Mathematics 31: 373–390.Google Scholar
35.Steutel, F.W. & van Harn, K. (1979). Discrete analogues of self-decomposability and stability. Annals of Probability 7: 893–899.Google Scholar
36.Steutel, F.W. & van Harn, K. (2003) Infinite divisibility of probability distributions on the real line. New York, Basel: Chapman & Hall/CRC Pure and Applied Mathematics.Google Scholar
37.Uchaikin, V.V. & Zolotarev, V.M. (1999). Chance and stability: Stable distributions and their applications. Utrecht: Walter de Gruyter, pp. 570.Google Scholar
38.van Harn, K., Steutel, F.W. & Vervaat, W. (1982). Self-decomposable discrete distributions and branching processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 61: 97–118.Google Scholar
39.Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Advances in Applied Probability 11: 750–783.Google Scholar