Electronic Journal of Probability

Fluctuations for mean-field interacting age-dependent Hawkes processes

Julien Chevallier

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The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes $n$ goes to $+\infty $) being granted by the study performed in [9], the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale $n^{-1/2}$) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differential equations driven by a Gaussian noise instead of Poisson (which occurs for the law of large numbers limit).

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 42, 49 pp.

Received: 7 November 2016
Accepted: 27 April 2017
First available in Project Euclid: 3 May 2017

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Primary: 60G55: Point processes 60F05: Central limit and other weak theorems 60G57: Random measures 60H15: Stochastic partial differential equations [See also 35R60] 92B20: Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx]

Hawkes process central limit theorem interacting particle systems stochastic partial differential equation neural network

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Chevallier, Julien. Fluctuations for mean-field interacting age-dependent Hawkes processes. Electron. J. Probab. 22 (2017), paper no. 42, 49 pp. doi:10.1214/17-EJP63. https://projecteuclid.org/euclid.ejp/1493777018

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  • [1] R. A. Adams and J. J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 2003.
  • [2] E. Bacry, K. Dayri, and J.-F. Muzy. Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data. The European Physical Journal B-Condensed Matter and Complex Systems, 85(5):1–12, 2012.
  • [3] E. Bacry, S. Delattre, M. Hoffmann, and J. F. Muzy. Scaling limits for Hawkes processes and application to financial statistics, Feb. 2012.
  • [4] P. Bao, H.-W. Shen, X. Jin, and X.-Q. Cheng. Modeling and predicting popularity dynamics of microblogs using self-excited Hawkes processes. In Proceedings of the 24th International Conference on World Wide Web, pages 9–10. ACM, 2015.
  • [5] P. Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. A Wiley-Interscience Publication.
  • [6] P. Brémaud. Point processes and queues. Springer-Verlag, New York, 1981. Martingale dynamics, Springer Series in Statistics.
  • [7] N. Brunel and V. Hakim. Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural computation, 11(7):1621–1671, 1999.
  • [8] M. A. Buice and C. C. Chow. Dynamic finite size effects in spiking neural networks. PLoS Comput Biol, 9(1):e1002872, 2013.
  • [9] J. Chevallier. Mean-field limit of generalized Hawkes processes. (to appear in Stochastic Processes and their Applications) arXiv preprint arXiv:1510.05620, 2015.
  • [10] J. Chevallier. Modelling large neural networks via Hawkes processes. Theses, Université Côte d’Azur, Sept. 2016.
  • [11] J. Chevallier, M. J. Cáceres, M. Doumic, and P. Reynaud-Bouret. Microscopic approach of a time elapsed neural model. Mathematical Models and Methods in Applied Sciences, 25(14):2669–2719, 2015.
  • [12] R. Crane and D. Sornette. Robust dynamic classes revealed by measuring the response function of a social system. Proceedings of the National Academy of Sciences, 105(41):15649–15653, 2008.
  • [13] G. Dumont et al. Private communication about ongoing work.
  • [14] G. Dumont, J. Henry, and C. O. Tarniceriu. Theoretical connections between mathematical neuronal models corresponding to different expressions of noise. Journal of Theoretical Biology, 406:31–41, 2016.
  • [15] O. Faugeras and J. Maclaurin. Asymptotic description of stochastic neural networks. i. existence of a large deviation principle. Comptes Rendus Mathematique, 352(10):841–846, 2014.
  • [16] R. Ferland, X. Fernique, and G. Giroux. Compactness of the fluctuations associated with some generalized nonlinear boltzmann equations. Canadian journal of mathematics, 44(6):1192–1205, 1992.
  • [17] B. Fernandez and S. Méléard. A Hilbertian approach for fluctuations on the McKean-Vlasov model. Stochastic Process. Appl., 71(1):33–53, 1997.
  • [18] W. Gerstner and W. M. Kistler. Spiking neuron models: Single neurons, populations, plasticity. Cambridge university press, 2002.
  • [19] R. D. Gill, N. Keiding, and P. K. Andersen. Statistical models based on counting processes. Springer, 1997.
  • [20] G. Gusto and S. Schbath. FADO: A Statistical Method to Detect Favored or Avoided Distances between Occurrences of Motifs using the Hawkes’ Model. Statistical Applications in Genetics and Molecular Biology, 4(1), 2005.
  • [21] A. G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):83–90, 1971.
  • [22] J. Jacod and A. N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2003.
  • [23] A. Joffe and M. Métivier. Weak convergence of sequences of semimartingales with applications to multitype branching processes. Advances in Applied Probability, pages 20–65, 1986.
  • [24] B. Jourdain and S. Méléard. Propagation of chaos and fluctuations for a moderate model with smooth initial data. In Annales de l’IHP Probabilités et statistiques, volume 34, pages 727–766, 1998.
  • [25] Y. Y. Kagan. Statistical distributions of earthquake numbers: consequence of branching process. Geophysical Journal International, 180(3):1313–1328, 2010.
  • [26] T. J. Liniger. Multivariate Hawkes processes. PhD thesis, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 18403, 2009, 2009.
  • [27] E. Luçon and W. Stannat. Transition from gaussian to non-gaussian fluctuations for mean-field diffusions in spatial interaction. The Annals of Applied Probability, 26(6):3840–3909, 2016.
  • [28] M. Mattia and P. Del Giudice. Finite-size dynamics of inhibitory and excitatory interacting spiking neurons. Physical Review E, 70(5):052903, 2004.
  • [29] S. Méléard. Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. In Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), volume 1627 of Lecture Notes in Math., pages 42–95. Springer, Berlin, 1996.
  • [30] S. Méléard. Convergence of the fluctuations for interacting diffusions with jumps associated with boltzmann equations. Stochastics: An International Journal of Probability and Stochastic Processes, 63(3-4):195–225, 1998.
  • [31] F. Merlevède and M. Peligrad. Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples. The Annals of Probability, 41(2):914–960, 2013.
  • [32] G. O. Mohler, M. B. Short, P. J. Brantingham, F. P. Schoenberg, and G. E. Tita. Self-exciting point process modeling of crime. Journal of the American Statistical Association, 106(493), 2011.
  • [33] Y. Ogata. Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50(2):379–402, 1998.
  • [34] K. Pakdaman, B. Perthame, and D. Salort. Dynamics of a structured neuron population. Nonlinearity, 23(1):55, 2010.
  • [35] K. Pakdaman, B. Perthame, and D. Salort. Relaxation and self-sustained oscillations in the time elapsed neuron network model. SIAM Journal on Applied Mathematics, 73(3):1260–1279, 2013.
  • [36] R. Rebolledo. Central limit theorems for local martingales. Z. Wahrsch. Verw. Gebiete, 51(3):269–286, 1980.
  • [37] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion (Grundlehren der mathematischen Wissenschaften). Springer-Verlag, 3rd edition, 1999.
  • [38] P. Reynaud-Bouret and S. Schbath. Adaptive estimation for Hawkes processes; application to genome analysis. The Annals of Statistics, 38(5):2781–2822, 2010.
  • [39] M. G. Riedler, M. Thieullen, and G. Wainrib. Limit theorems for infinite-dimensional piecewise deterministic markov processes. applications to stochastic excitable membrane models. Electron. J. probab, 17(55):1–48, 2012.
  • [40] G. R. Shorack and J. A. Wellner. Empirical processes with applications to statistics, volume 59. Siam, 2009.
  • [41] A.-S. Sznitman. Topics in propagation of chaos. In École d’Été de Probabilités de Saint-Flour XIX—1989, volume 1464 of Lecture Notes in Math., pages 165–251. Springer, Berlin, 1991.
  • [42] V. C. Tran. Modèles particulaires stochastiques pour des problèmes d’évolution adaptative et pour l’approximation de solutions statistiques. PhD thesis, Université de Nanterre-Paris X, 2006.
  • [43] C. Tuleau-Malot, A. Rouis, F. Grammont, and P. Reynaud-Bouret. Multiple Tests Based on a Gaussian Approximation of the Unitary Events Method with delayed coincidence count. appearing in Neural Computation, 26:7, 2014.
  • [44] G. Wainrib. Randomness in neurons: a multiscale probabilistic analysis. PhD thesis, PhD thesis, 2010.
  • [45] K. Yosida. Functional analysis, volume 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin-New York, sixth edition, 1980.
  • [46] L. Zhu. Nonlinear Hawkes Processes. PhD thesis, New York University, 2013.