## Electronic Journal of Statistics

### Statistical inference versus mean field limit for Hawkes processes

#### Abstract

We consider a population of $N$ individuals, of which we observe the number of actions until time $t$. For each couple of individuals $(i,j)$, $j$ may or not influence $i$, which we model by i.i.d. Bernoulli$(p)$-random variables, for some unknown parameter $p\in(0,1]$. Each individual acts autonomously at some unknown rate $\mu>0$ and acts by mimetism at some rate proportional to the sum of some function $\varphi$ of the ages of the actions of the individuals which influence him. The function $\varphi$ is unknown but assumed, roughly, to be decreasing and with fast decay. The goal of this paper is to estimate $p$, which is the main characteristic of the graph of interactions, in the asymptotic $N\to\infty$, $t\to \infty$. The main issue is that the mean field limit (as $N\to\infty$) of this model is unidentifiable, in that it only depends on the parameters $\mu$ and $p\varphi$. Fortunately, this mean field limit is not valid for large times. We distinguish the subcritical case, where, roughly, the mean number $m_{t}$ of actions per individual increases linearly and the supercritical case, where $m_{t}$ increases exponentially. Although the nuisance parameter $\varphi$ is non-parametric, we are able, in both cases, to estimate $p$ without estimating $\varphi$ in a nonparametric way, with a precision of order $N^{-1/2}+N^{1/2}m_{t}^{-1}$, up to some arbitrarily small loss. We explain, using a Gaussian toy model, the reason why this rate of convergence might be (almost) optimal.

#### Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 1223-1295.

Dates
First available in Project Euclid: 12 May 2016

https://projecteuclid.org/euclid.ejs/1463068346

Digital Object Identifier
doi:10.1214/16-EJS1142

Mathematical Reviews number (MathSciNet)
MR3499526

Zentralblatt MATH identifier
1343.62050

#### Citation

Delattre, Sylvain; Fournier, Nicolas. Statistical inference versus mean field limit for Hawkes processes. Electron. J. Statist. 10 (2016), no. 1, 1223--1295. doi:10.1214/16-EJS1142. https://projecteuclid.org/euclid.ejs/1463068346

#### References

• [1] Y. Aït-Sahalia, J. Cacho-Diaz, R. J. A. Laeven, Modeling financial contagion using mutually exciting jump processes, to appear in the Journal of Financial Economics.
• [2] E. Bacry, S. Delattre, M. Hoffmann, J. F. Muzy, Modeling microstructure noise with mutually exciting point processes, Quantitative Finance 13 (2013), 65–77.
• [3] E. Bacry, S. Delattre, M. Hoffmann, J. F. Muzy, Some limit theorems for Hawkes processes and applications to financial statistics, Stoch. Processes Appl. 123 (2013), 2475–2499.
• [4] E. Bacry and J. F. Muzy, Hawkes model for price and trades high-frequency dynamics, Quantitative Finance 14 (2014), 1147–1166.
• [5] E. Bacry and J. F. Muzy, Second order statistics characterization of Hawkes processes and non-parametric estimation, arXiv :1401.0903.
• [6] L. Bauwens, N. Hautsch, Modeling financial high frequency data using point processes, ser. In T. Mikosch, J.-P. Kreiss, R. A. Davis, and T. G. Andersen, editors, Handbook of Financial Time Series. Springer, 2009.
• [7] G. Birkhoff, Extensions of Jentzsch’s theorem, Trans. Am. Math. Soc. 85 (1957), 219–227.
• [8] C. Blundell, K. A. Heller, J. F. Beck, Modeling reciprocating relationships with Hawkes processes, Neural Information Processing Systems 2012.
• [9] P. Brémaud, L. Massoulié, Stability of nonlinear Hawkes processes, Ann. Probab. 24 (1996), 1563–1588.
• [10] P. Brémaud, G. Nappo, G. L. Torrisi, Rate of convergence to equilibrium of marked Hawkes processes, J. Appl. Probab. 39 (2002), 123–136.
• [11] R. Cavazos-Cadena, An alternative derivation of Birkhoff’s formula for the contraction coefficient of a positive matrix, Linear Algebra Appl. 375 (2003), 291–297.
• [12] D. J. Daley, D. Vere-Jones, An introduction to the theory of point processes, Vol. I. Probability and its Applications. Springer-Verlag, second edition, 2003.
• [13] S. Delattre, N. Fournier, M. Hoffmann, Hawkes processes on large networks, to appear in Ann. Appl. Probab.
• [14] W. Feller, On the integral equation of renewal theory, Ann. Math. Statistics 12 (1941), 243–267.
• [15] S. Grün, M. Diedsmann, A. M. Aertsen, Unitary events analysis, in Analysis of parallel spike trains, S. Grün and S. Rotter, Springer series in computational neurosciences, 2010.
• [16] N. R. Hansen, P. Reynaud-Bouret, V. Rivoirard, Lasso and probabilistic inequalities for multivariate point processes, to appear in Bernoulli.
• [17] A. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika 58 (1971), 83–90.
• [18] A. Hawkes, D. Oakes, A cluster process representation of a self-exciting process, J. Appl. Probability 11 (1974), 493–503.
• [19] A. Helmstetter and D. Sornette, Subcritical and supercritical regimes in epidemic models of earthquake aftershocks, Journal of geophysical research, 107 (2002), 2237.
• [20] P. Hewlett, Clustering of order arrivals, price impact and trade path optimisation, In Workshop on Financial Modeling with Jump processes. Ecole Polytechnique, 2006.
• [21] W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.
• [22] J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, Second edition. Springer-Verlag, 2003.
• [23] Y. Y. Kagan, Statistical distributions of earthquake numers: consequence of branching process, Geophysical Journal International 180 (2010), 1313–1328.
• [24] L. Massoulié, Stability results for a general class of interacting point processes dynamics, and applications, Stochastic Process. Appl. 75 (1998), 1–30.
• [25] Y. Ogata, The asymptotic behaviour of maximum likelihood estimators for stationary point processes, Ann. Instit. Math. Statist. 30 (1978), 243–261.
• [26] Y. Ogata, Seismicity analysis through point-process modeling: a review, Pure and Applied Geophysics 155 (1999), 471–507.
• [27] M. Okatan, M. A. Wilson, E. N. Brown, Analyzing functional connectivity using a network likelihood model of ensemble neural spiking activity, Neural Computation 17 (2005), 1927-1961.
• [28] J. W. Pillow, J. Shlens, L. Paninski, A. Scher, A. M. Litke, E. J. Chichilnisky, E. P. Simoncelli, Spatio-temporal correlations and visual signalling in a complete neuronal population, Nature 454 (2008), 995– 999.
• [29] J. G. Rasmussen, Bayesian inference for Hawkes processes, Methodol. Comput. Appl. Probab. 15 (2013), 623–642.
• [30] P. Reynaud-Bouret and S. Schbath, Adaptive estimation for Hawkes processes: application to genome analysis, Ann. Statist. 38 (2010), 2781–2822.
• [31] P. Reynaud-Bouret, V. Rivoirard, F. Grammont, C. Tuleau-Malot, Goodness-of-fit tests and nonparametric adaptive estimation for spike train analysis, Journal of Math. Neuroscience 4:3 (2014).
• [32] P. Reynaud-Bouret, V. Rivoirard, C. Tuleau-Malot, Inference of functional connectivity in Neurosciences via Hawkes processes, 1st IEEE Global Conference on Signal and Information Processing, 2013.
• [33] A. S. Sznitman, Topics in propagation of chaos, Ecole d’Été de Probabilités de Saint-Flour XIX -1989, Vol. 1464 of Lecture Notes in Math. Springer, 1991, 165–251.
• [34] K. Zhou, H. Zha, L. Song, Learning triggering kernels for multi-dimensional Hawkes processes, Proceedings of the 30th International Conference on Machine Learning, 2013.
• [35] L. Zhu, Central limit theorem for nonlinear Hawkes processes, J. App. Probab. 50 (2013), 760–771.
• [36] L. Zhu, Large deviations for Markovian nonlinear Hawkes processes, Ann. App. Probab. 25 (2015), 548–581.