E. Bacry, S. Delattre, M. Hoffmann, and J. F. Muzy, Modelling microstructure noise with mutually exciting point processes, Quant. Finance, vol.13, issue.1, pp.65-77, 2013.
DOI : 10.1080/14697688.2011.647054
URL : https://hal.archives-ouvertes.fr/hal-00830987

E. Bacry, S. Delattre, M. Hoffmann, and J. F. Muzy, Some limit theorems for Hawkes processes and application to financial statistics, Stochastic Process. Appl, vol.123, issue.7, pp.2475-2499, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00845004

E. Bacry and J. Muzy, First-and second-order statistics characterization of Hawkes processes and non-parametric estimation, IEEE Trans. Inform. Theory, vol.62, issue.4, pp.2184-2202, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01313834

J. Barral, Y. Hu, and T. Madaule, The minimum of a branching random walk outside the boundary case, Bernoulli, vol.24, issue.2, pp.801-841, 2018.

H. C. Berbee, Random walks with stationary increments and renewal theory, Mathematical Centre Tracts. Mathematisch Centrum, vol.112, 1979.

P. Brémaud and L. Massoulié, Stability of nonlinear Hawkes processes, Ann. Probab, vol.24, issue.3, pp.1563-1588, 1996.

J. Chevallier, Mean-field limit of generalized Hawkes processes, Stochastic Process. Appl, vol.127, issue.12, pp.3870-3912, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01217407

J. Chevallier, M. J. Cáceres, M. Doumic, and P. Reynaud-bouret, Microscopic approach of a time elapsed neural model, Math. Models Methods Appl. Sci, vol.25, issue.14, pp.2669-2719, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01159215

M. Costa, C. Graham, L. Marsalle, and V. Tran, Renewal in Hawkes processes with self-excitation and inhibition, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01683954

D. J. Daley and D. Vere-jones, An introduction to the theory of point processes, Probability and its Applications, 2003.

D. J. Daley and D. Vere-jones, An introduction to the theory of point processes, II. Probability and its Applications

S. Delattre and N. Fournier, Statistical inference versus mean field limit for Hawkes processes, Electron. J. Stat, vol.10, issue.1, pp.1223-1295, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01367752

S. Delattre, N. Fournier, and M. Hoffmann, Hawkes processes on large networks, Ann. Appl. Probab, vol.26, issue.1, pp.216-261, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01102806

S. Ditlevsen and E. Löcherbach, Multi-class oscillating systems of interacting neurons, Stochastic Process. Appl, vol.127, issue.6, pp.1840-1869, 2017.

A. Duarte, E. Löcherbach, and G. Ost, Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels, 2016.

F. Guillemin and A. Simonian, Transient characteristics of an M/M/? system, Adv. in Appl. Probab, vol.27, issue.3, pp.862-888, 1995.
URL : https://hal.archives-ouvertes.fr/inria-00074289

A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika, vol.58, pp.83-90, 1971.

A. G. Hawkes and L. Adamopoulos, Cluster models for earthquakes: Regional comparisons, Bull. Int. Statist. Inst, vol.45, pp.454-461, 1973.

A. G. Hawkes and D. Oakes, A cluster process representation of a self-exciting process, J. Appl. Probability, vol.11, pp.493-503, 1974.

Y. Hu, How big is the minimum of a branching random walk?, Ann. Inst. Henri Poincaré Probab. Stat, vol.52, issue.1, pp.233-260, 2016.
URL : https://hal.archives-ouvertes.fr/hal-00826652

T. Jaisson and M. Rosenbaum, Limit theorems for nearly unstable Hawkes processes, Ann. Appl. Probab, vol.25, issue.2, pp.600-631, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01138784

T. Jaisson and M. Rosenbaum, Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes, Ann. Appl. Probab, vol.26, issue.5, pp.2860-2882, 2016.

N. N. Lebedev, Special functions and their applications, 1965.

B. Mallein, Genealogy of the extremal process of the branching random walk, ALEA Lat. Am. J. Probab. Math. Stat, vol.15, issue.2, pp.1065-1087, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01326628

P. Massart, Lectures from the 33rd Summer School on Probability Theory, Lecture Notes in Mathematics, vol.1896, 2003.

J. Möller and J. G. Rasmussen, Perfect simulation of Hawkes processes, Adv. in Appl. Probab, vol.37, issue.3, pp.629-646, 2005.

Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes, Journal of American Statistical Association, vol.83, issue.401, pp.9-27, 1988.

P. Reynaud-bouret, V. Rivoirard, and C. Tuleau-malot, Inference of functional connectivity in Neurosciences via Hawkes processes, 1st IEEE Global Conference on Signal and Information Processing, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00866823

P. Reynaud-bouret and E. Roy, Some non asymptotic tail estimates for Hawkes processes, Bull. Belg. Math. Soc. Simon Stevin, vol.13, issue.5, pp.883-896, 2006.

P. Reynaud-bouret and S. Schbath, Adaptive estimation for Hawkes processes; application to genome analysis, Ann. Statist, vol.38, issue.5, pp.2781-2822, 2010.
DOI : 10.1214/10-aos806
URL : https://hal.archives-ouvertes.fr/hal-00863958

P. Robert, Stochastic networks and queues, Stochastic Modelling and Applied Probability, vol.52, 2003.
DOI : 10.1007/978-3-662-13052-0

W. Rudin, Real and complex analysis, 1987.

Z. Shi, Branching random walks, Lecture Notes in Mathematics, vol.2151, 2012.
DOI : 10.1007/978-3-319-25372-5
URL : https://hal.archives-ouvertes.fr/hal-01275563

L. Takács, On a probability problem arising in the theory of counters, Proc. Cambridge Philos. Soc, vol.52, pp.488-498, 1956.

L. Takács, University Texts in the Mathematical Sciences, 1962.

H. Thorisson, Coupling, stationarity, and regeneration. Probability and its Applications, 2000.

D. Widder, The Laplace Transform, Princeton Mathematical Series, issue.6, 1941.