Limit theorems for discrete Hawkes processes. (English) Zbl 1315.60058
Summary: We consider a discrete time Hawkes process which is a class of \(g\)-functions. The limit theorems for continuous time Hawkes processes are well known and studied by many authors. In this paper, we study the limit theorems for discrete time Hawkes processes. In particular, we obtain the law of large numbers, the central limit theorem and the invariance principle.
MSC:
60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
60F05 | Central limit and other weak theorems |
60F15 | Strong limit theorems |
60F17 | Functional limit theorems; invariance principles |
Keywords:
discrete Hawkes processes; self-exciting processes; central limit theorem; law of large numbers; invariance principleReferences:
[1] | Bacry, E.; Delattre, S.; Hoffmann, M.; Muzy, J. F., Scaling limits for Hawkes processes and application to financial statistics, Stochastic Process. Appl., 123, 2475-2499 (2012) · Zbl 1292.60032 |
[2] | Berbee, H., Chains with infinite connections: uniquenss and Markov representation, Probab. Theory Related Fields, 76, 243-253 (1987) · Zbl 0611.60059 |
[3] | Bordenave, C.; Torrisi, G. L., Large deviations of Poisson cluster processes, Stoch. Models, 23, 593-625 (2007) · Zbl 1152.60316 |
[4] | Bramson, M.; Kalikow, S., Nonuniqueness in \(g\)-functions, Israel J. Math., 84, 153-160 (1993) · Zbl 0786.60043 |
[5] | Brémaud, P.; Massoulié, L., Stability of nonlinear Hawkes processes, Ann. Probab., 24, 1563-1588 (1996) · Zbl 0870.60043 |
[6] | Daley, D. J.; Vere-Jones, D., An Introduction to the Theory of Point Processes, Vols. I and II (2003), Springer · Zbl 1026.60061 |
[7] | Dassios, A.; Zhao, H., A dynamic contagion process, Adv. in Appl. Probab., 43, 814-846 (2011) · Zbl 1230.60089 |
[8] | Doeblin, W.; Fortet, R., Sur les chaines a liaisons completes, Bull. Soc. Math. France, 65, 132-148 (1937) · JFM 63.1077.05 |
[9] | Errais, E.; Giesecke, K.; Goldberg, L., Affine point processes and portfolio credit risk, SIAM J. Financial Math., 1, 642-665 (2010) · Zbl 1200.91296 |
[10] | Fortuin, C.; Kasteleyn, P. W.; Ginibre, J., Correlation inequalities on some partially ordered sets, Comm. Math. Phys., 22, 89-103 (1971) · Zbl 0346.06011 |
[11] | Hawkes, A. G., Spectra of some self-exciting and mutually exciting point process, Biometrika, 58, 83-90 (1971) · Zbl 0219.60029 |
[13] | Keane, M., Strongly mixing \(g\)-measures, Invent. Math., 16, 309-324 (1972) · Zbl 0241.28014 |
[14] | Meester, R.; Roy, R., Continuum percolation, (Cambridge Tracts in Mathematics (1996), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0858.60092 |
[16] | Oliveira, P. E., Asymptotics for Associated Random Variables (2012), Springer · Zbl 1249.62001 |
[17] | Zhu, L., Central limit theorem for nonlinear Hawkes processes, J. Appl. Probab., 50, 760-771 (2013) · Zbl 1306.60015 |
[18] | Zhu, L., Moderate deviations for Hawkes processes, Statist. Probab. Lett., 83, 885-890 (2013) · Zbl 1266.60090 |
[19] | Zhu, L., Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims, Insurance Math. Econom., 53, 544-550 (2013) · Zbl 1290.91107 |
[20] | Zhu, L., Process-level large deviations for nonlinear Hawkes point processes, Ann. Inst. Henri Poincaré, 50, 845-871 (2014) · Zbl 1296.60129 |
[21] | Zhu, L., Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps, J. Appl. Probab., 51, 699-712 (2014) · Zbl 1307.60033 |
[22] | Zhu, L., Large deviations for Markovian nonlinear Hawkes processes, Ann. Appl. Probab. (2014), in press |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.