Large deviations of poisson cluster processes

Stochastic Models

Volumn 23, Issue 4, 2007, Pages 593-625

  Bordenave, Charles ^a,c   Torrisi, Giovanni Luca ^b  

^a INRIA   (France)

^b CNR   (Italy)

^c ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

Hawkes processes; Large deviations; Poisson cluster processes; Poisson processes

Indexed keywords

CLUSTER ANALYSIS;

HAWKES PROCESSES; LARGE DEVIATIONS; POISSON CLUSTER PROCESSES;

POISSON DISTRIBUTION;

EID: 35748939067     PISSN: 15326349     EISSN: 15324214     Source Type: Journal    
DOI: 10.1080/15326340701645959     Document Type: Article

References (32)

1. Borovkov, A.A. Boundary values problems for random walks and large deviations for function spaces. Theory Probab. Appl. 1967, 12, 575-595.
2. Brémaud, P.; Massoulié, L. Stability of nonlinear Hawkes processes. Ann. Prob. 1996, 24, 1563-1588.
3. Brémaud, P.; Massoulié, L.; Ridolfi, A. Power spectra of random spike fields and related processes. Adv. Appl. Prob. 2005, 37, 1116-1146.
4. Brémaud, P.; Nappo, G.; Torrisi, G.L. Rate of convergence to equilibrium of marked Hawkes processes. J. Appl. Prob. 2002, 39, 123-136.
5. Brix, A.; Chadoeuf, J. Spatio-temporal modeling of weeds by shot-noise G Cox processes. Biometrical J. 2002, 44, 83-99.
6. Chavez-Demoulin, V.; Davison, A.C.; Mc Neil, A.J. Estimating value-at-risk: a point process approach. Quantitative Finance. 2005, 5, 227-234.
7. Daley, DJ.; Vere-Jones, D. An Introduction to the Theory of Point Processes, 2nd Ed.; Springer: New York, 2003.
8. de Acosta, A. Large deviations for vector valued Levy processes. Stochastic Process. Appl. 1994, 51, 75-115.
9. Dembo, A.; Zeitouni, O. Large Deviations Techniques and Applications, 2nd Ed.; Springer: New York, 1998.
10. Ganesh, A.; Macci, C.; Torrisi, G.L. Sample path large deviations principles for Poisson shot noise processes and applications. Electron. J. Probab. 2005, 10, 1026-1043.
11. Gusto, G.; Schbath, S. F.A.D.O.: A statistical method to detect favored or avoided distances between occurrences of motifs using the Hawkes model. Stat. Appl. Genet. Mol. Biol. 2005, 4, Article 24.
12. Hawkes, A.G. Spectra of some self-exciting and mutually exciting point processes. Biometrika 1971, 58, 83-90.
13. Hawkes, A.G. Point spectra of some mutually exciting point processes. J. Roy. Statist. Soc. Ser. B 1971, 33, 438-443.
14. Hawkes, A.G.; Oakes, D. A cluster representation of a self-exciting process. J. Appl. Prob. 1974, 11, 493-503.
15. Jagers, P. Branching Processes with Biological Applications; John Wiley: London, 1975.
16. Jonnson, D.H. Point process models of single-neuron discharges. J. Computational Neuroscience 1996, 3, 275-299.
17. Kerstan, J. Teilprozesse poissonscher prozesse. Transactions of the Third Prague Conference on Information Theory, Statistical Decision Functions, Random Processes 1964; 377-403.
18. Massoulié, L. Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Proc. Appl. 1998, 75, 1-30.
19. Møller, J. A comparison of spatial point process models in epidemiological applications. In Highly Structured Stochastic Systems; Green, P.J.; Hjort, N.L.; Richardson, S., Eds; Oxford University Press, 2003; 264-268.
20. Møller, J. Shot noise Cox processes. Adv. Appl. Prob. 2003, 35, 614-640.
21. Møller, J.; Rasmussen, J.G. Perfect simulation of Hawkes processes. Adv. Appl. Prob. 2005, 37, 629-646.
22. Møller, J.; Torrisi, G.L. Generalised shot noise Cox processes. Adv. Appl. Prob. 2005, 37, 48-74.
23. Møller, J.; Torrisi, G.L. The pair correlation function of spatial Hawkes processes. Stat. Probab. Lett. 2007, 77, 995-1003.
24. Møller, J.; Waagepetersen, R.S. Statistical Inference and Simulation for Spatial Point Processes; Chapman and Hall: Boca Raton, 2004.
25. Neyman, J.; Scott, E.L. Statistical approach to problems of cosmology. J. R. Statist. Soc. B 1958, 20, 1-43.
26. Ogata, Y. Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 1988, 83, 9-27.
27. Ogata, Y. Space-time point process model for earthquake occurrences. Ann. Inst. Statist. Math. 1998, 50, 379-402.
28. Ogata, Y.; Akaike, H. On linear intensity models for mixed doubly stochastic poisson and selfexciting point processes. J. R. Statist. Soc. B 1982, 44, 102-107.
29. Reynaud-Bouret, P.; Roy, E. Some non asymptotic tail estimates for Hawkes processes. Bull. Belg. Math. Soc. Simon Stevin 2007, 13, 883-896.
30. Stabile, G.; Torrisi, G.L. Risk processes with non-stationary Hawkes arrivals. Submitted. 2007.
31. Torrisi, G.L. A class of interacting marked point processes: rate of convergence to equilibrium. J. Appl. Prob. 2002, 39, 137-161.
32. Vere-Jones, D.; Ozaki, T. Some examples of statistical estimation applied to earthquake data. Ann. Inst. Statist. Math. 1982, 34, 189-207.