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On a Two-Parameter Yule-Simon Distribution

Part of the Progress in Probability book series (PRPR,volume 78)

Abstract

We extend the classical one-parameter Yule-Simon law to a version depending on two parameters, which in part appeared in Bertoin (J Stat Phys 176(3):679–691, 2019) in the context of a preferential attachment algorithm with fading memory. By making the link to a general branching process with age-dependent reproduction rate, we study the tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already initiated in Bertoin (J Stat Phys 176(3):679–691, 2019). Finally, by superposing mutations to the branching process, we propose a model which leads to the two-parameter range of the Yule-Simon law, generalizing thereby the work of Simon (Biometrika 42(3/4):425–440, 1955) on limiting word frequencies.

Keywords

  • Yule-Simon model
  • Crump-Mode-Jagers branching process
  • Population model with neutral mutations
  • Heavy tail distribution
  • Preferential attachment with fading memory

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Notes

  1. 1.

    Although the probability \(\mathbb {P}(X_{\theta , \rho }=1)\) can easily be computed in terms of an incomplete Gamma function, the calculations needed to determine \(\mathbb {P}(X_{\theta , \rho }=k)\) for k ≥ 2 become soon intractable.

References

  1. Bertoin, J.: A version of Herbert A. Simon’s model with slowly fading memory and its connections to branching processes. J. Stat. Phys. 176(3), 679–691 (2019)

    MATH  Google Scholar 

  2. Bingham, N.H., Doney, R.A.: Asymptotic properties of supercritical branching processes. II. Crump-Mode and Jirina processes. Adv. Appl. Probab. 7, 66–82 (1975)

    CrossRef  Google Scholar 

  3. Caballero, M.E., Lambert, A., Uribe Bravo, G.: Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surv. 6, 62–89 (2009)

    MathSciNet  CrossRef  Google Scholar 

  4. Doney, R.A.: A limit theorem for a class of supercritical branching processes. J. Appl. Probab. 9, 707–724 (1972)

    MathSciNet  CrossRef  Google Scholar 

  5. Doney, R.A.: On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Relat. Fields 81(2), 239–246 (1989)

    MathSciNet  CrossRef  Google Scholar 

  6. Doney, R.A.: Local behaviour of first passage probabilities. Probab. Theory Relat. Fields 152(3–4), 559–588 (2012)

    MathSciNet  CrossRef  Google Scholar 

  7. Doney, R.A., Rivero, V.: Asymptotic behaviour of first passage time distributions for Lévy processes. Probab. Theory Relat. Fields 157(1–2), 1–45 (2013)

    CrossRef  Google Scholar 

  8. Drmota, M.: Random Trees: An Interplay Between Combinatorics and Probability, 1st edn. Springer Publishing Company, Incorporated, Wien/London (2009)

    CrossRef  Google Scholar 

  9. Ethier, S.N., Kurtz, T.G.: Markov Processes. Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)

    Google Scholar 

  10. Jiřina, M.: Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8(83), 292–313 (1958)

    MathSciNet  CrossRef  Google Scholar 

  11. Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications. Introductory Lectures. Universitext, 2nd edn. Springer, Heidelberg (2014)

    Google Scholar 

  12. Lansky, P., Polito, F., Sacerdote, L.: The role of detachment of in-links in scale-free networks. J. Phys. A: Math. Theor. 47(34), 345002 (2014)

    CrossRef  Google Scholar 

  13. Lansky, P., Polito, F., Sacerdote, L.: Generalized nonlinear Yule models. J. Stat. Phys. 165(3), 661–679 (2016)

    MathSciNet  CrossRef  Google Scholar 

  14. Mandelbrot, B.: A note on a class of skew distribution functions: Analysis and critique of a paper by H. A. Simon. Inf. Control 2(1), 90–99 (1959)

    MathSciNet  CrossRef  Google Scholar 

  15. Nerman, O.: On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrsch. Verw. Gebiete 57(3), 365–395 (1981)

    MathSciNet  CrossRef  Google Scholar 

  16. Pitman, J.: Combinatorial Stochastic Processes. Lecture Notes in Mathematics, vol. 1875. Springer, Berlin (2006). Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, 7–24 July 2002, With a foreword by Jean Picard

    Google Scholar 

  17. Polito, F.: Studies on generalized Yule models. Modern Stoch.: Theory Appl. 6(1), 41–55 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Portnoy, S.: Probability bounds for first exits through moving boundaries. Ann. Probab. 6(1), 106–117 (1978)

    MathSciNet  CrossRef  Google Scholar 

  19. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales. Vol. 1. Foundations. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, 2nd edn. Wiley, Chichester (1994)

    Google Scholar 

  20. Simon, H.A.: On a class of skew distribution functions. Biometrika 42(3/4), 425–440 (1955)

    MathSciNet  CrossRef  Google Scholar 

  21. Simon, H.A.: Some further notes on a class of skew distribution functions. Inf. Control 3(1), 80–88 (1960)

    MathSciNet  CrossRef  Google Scholar 

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Baur, E., Bertoin, J. (2021). On a Two-Parameter Yule-Simon Distribution. In: Chaumont, L., Kyprianou, A.E. (eds) A Lifetime of Excursions Through Random Walks and Lévy Processes. Progress in Probability, vol 78. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83309-1_4

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