Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-12T11:34:48.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Renewal in Hawkes processes with self-excitation and inhibition

Part of: Stochastic processes Limit theorems Special processes Integral transforms, operational calculus

Published online by Cambridge University Press:  24 September 2020

Manon Costa Open the ORCID record for Manon Costa [Opens in a new window] ,
Carl Graham ,
Laurence Marsalle  and
Viet Chi Tran
Manon Costa*
Affiliation:
Université Toulouse III
Carl Graham*
Affiliation:
École Polytechnique
Laurence Marsalle*
Affiliation:
Université de Lille
Viet Chi Tran*
Affiliation:
LAMA, Université Gustave Eiffel, UPEM, Université Paris-Est Créteil, CNRS
*
*Postal address: Institut de Mathématiques de Toulouse, UMR 5219; Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France. Email address: [email protected]
**Postal address: CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France.
***Postal adddress: Université de Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France.
***Postal adddress: Université de Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.

Keywords

Point processesself-excitationinhibitionergodic limit theoremsconcentration inequalitiesGalton–Watson treesM/G/∞ queues

MSC classification

Primary: 60G55: Point processes 60F99: None of the above, but in this section
Secondary: 60K05: Renewal theory 60K25: Queueing theory 44A10: Laplace transform
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 3 , September 2020 , pp. 879 - 915
Check for updates
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Modeling microstructure noise with mutually exciting point processes. Quant. Finance 13, 65–77.CrossRefGoogle Scholar
Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Some limit theorems for Hawkes processes and application to financial statistics. Stoch. Proc. Appl. 123, 2475–2499.CrossRefGoogle Scholar
Bacry, E. and Muzy, J. F. (2016). First- and second-order statistics characterization of Hawkes processes and non-parametric estimation. IEEE Trans. Inf. Theory 62, 2184–2202.CrossRefGoogle Scholar
Berbee, H. C. P. (1979). Random Walks with Stationary Increments and Renewal Theory (Mathematical Centre Tracts 112). Mathematisch Centrum, Amsterdam.Google Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.Google Scholar
Brémaud, P., Nappo, G. and Torrisi, G. L. (2002). Rate of convergence to equilibrium of marked Hawkes processes. J. Appl. Prob. 39, 123–136.CrossRefGoogle Scholar
Chen, S., Shojaie, A., Shea-Brown, E. and Witten, D. (2017). The multivariate Hawkes process in high dimensions: Beyond mutual excitation. Preprint. Available at https://arxiv.org/abs/1707.04928.Google Scholar
Chevallier, J. (2017). Mean-field limit of generalized Hawkes processes. Stoch. Proc. Appl. 127, 3870–3912.CrossRefGoogle Scholar
Chevallier, J., Caceres, M. J., Doumic, M. and Reynaud-Bouret, P. (2015). Microscopic approach of a time elapsed neural model. Math. Models Meth. Appl. Sci. 25, 2669–2719.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Delattre, S. and Fournier, N. (2016). Statistical inference versus mean-field limit for Hawkes processes. Electron. J. Statist. 10, 1223–1295.CrossRefGoogle Scholar
Delattre, S., Fournier, N. and Hoffmann, M. (2016). Hawkes processes on large networks. Ann. Appl. Prob. 26, 216–261.CrossRefGoogle Scholar
Ditlevsen, S. and Löcherbach, E. (2017). On oscillating systems of interacting neurons. Stoch. Proc. Appl. 127, 1840–1869.CrossRefGoogle Scholar
Duarte, A., Löcherbach, E. and Ost, G. (2016). Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels. Preprint. Available at https://arxiv.org/abs/1610.03300.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York–London–Sydney.Google Scholar
Hansen, N. R., Reynaud-Bouret, P. and Rivoirard, V. (2015). Lasso and probabilistic inequalities for multivariate point processes. Bernoulli 21, 83–143.CrossRefGoogle Scholar
Hawkes, A. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. and Adamopoulos, L. (1973). Cluster models for earthquakes: Regional comparisons. Bull. Internat. Statist. Inst. 45, 454–461.Google Scholar
Hawkes, A. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.CrossRefGoogle Scholar
Jaisson, T. and Rosenbaum, M. (2015). Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Prob. 25, 600–631.CrossRefGoogle Scholar
Jaisson, T. and Rosenbaum, M. (2016). Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes. Ann. Appl. Prob. 26, 2860–2882.CrossRefGoogle Scholar
Lambert, R. C. et al. (2018). Reconstructing the functional connectivity of multiple spike trains using Hawkes models. J. Neurosci. Meth. 297, 9–21.CrossRefGoogle ScholarPubMed
Massart, P. (2007). Concentration Inequalities and Model Selection: École d’Été de Probabilités de Saint-Flour XXXIII - 2003 (Lecture Notes Math. 1896). Springer, Berlin, Heidelberg.Google Scholar
Massoulié, L. (1998). Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Proc. Appl. 75, 1–30.CrossRefGoogle Scholar
Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83, 9–27.CrossRefGoogle Scholar
Raad, M. B. (2019). Renewal time points for Hawkes processes. Preprint. Available at https://arxiv.org/abs/1906.02036.Google Scholar
Reynaud-Bouret, P., Rivoirard, V. and Tuleau-Malot, C. (2013). Inference of functional connectivity in Neurosciences via Hawkes processes. In 1st IEEE Global Conference on Signal and Information Processing, Austin, Texas, pp. 317–320.CrossRefGoogle Scholar
Reynaud-Bouret, P. and Roy, E. (2006). Some non asymptotic tail estimates for Hawkes processes. Bull. Belg. Math. Soc. 13, 883–896.CrossRefGoogle Scholar
Reynaud-Bouret, P. and Schbath, S. (2010). Adaptive estimation for Hawkes processes; application to genome analysis. Ann. Statist. 38, 2781–2822.CrossRefGoogle Scholar
Robert, P. (2000). Réseaux et files d’attente: méthodes probabilistes (Mathématiques et Applications 35). Springer, Berlin, Heidelberg.Google Scholar
Rudin, W. (1987). Real and Complex Analysis, 3rd edn. McGraw-Hill, New York.Google Scholar
Takács, L. (1956). On a probability problem arising in the theory of counters. Proc. Camb. Phil. Soc. 52, 488–498.CrossRefGoogle Scholar
Takács, L. (1962). Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.CrossRefGoogle Scholar
Widder, D. V. (1941). The Laplace Transform (Princeton Mathematical Series 6). Princeton University Press.Google Scholar
Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Prob. 50, 760–771.CrossRefGoogle Scholar