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Functional central limit theorems and moderate deviations for Poisson cluster processes

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  24 September 2020

Fuqing Gao  and
Yujing Wang
Fuqing Gao*
Affiliation:
Wuhan University
Yujing Wang*
Affiliation:
Wuhan University
*
*Postal address: School of Mathematics and Statistics, Wuhan University, Wuhan430072, China. Email: fqgao@whu.edu.cn
**Postal address: School of Mathematics and Statistics, Wuhan University, Wuhan430072, China. Email: yujingwang@whu.edu.cn
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Abstract

In this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes.

Keywords

Poisson cluster processHawkes processmaximal inequalitycentral limit theoremmoderate deviationlarge deviation

MSC classification

Primary: 60F10: Large deviations 60F17: Functional limit theorems; invariance principles 60G55: Point processes

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 3 , September 2020 , pp. 916 - 941
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Copyright
© Applied Probability Trust 2020

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References

Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J.-F. (2013). Scaling limits for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123, 2475–2499.CrossRefGoogle Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley & Sons, New York.Google Scholar
Billingsley, P. (1986). Probability and Measure. John Wiley & Sons, New York.Google Scholar
Bogachev, L. and Daletskii, A. (2009). Poisson cluster measures: Quasi-invariance, integration by parts and equilibrium stochastic dynamics. J. Funct. Anal. 256, 432–478.CrossRefGoogle Scholar
Bordenave, C. and Torrisi, G. L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23, 593–625.CrossRefGoogle Scholar
Chun, Y. J., Hasna, M. O. and Ghrayeb, A. (2015). Modeling heterogeneous cellular networks interference using Poisson cluster processes. IEEE J. Sel. Areas Commun. 33, 2182–2195.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. 1, 2nd edn. Springer, New York.Google Scholar
Delattre, S., Fournier, N. and Hoffmann, M. (2016). Hawkes processes on large networks. Ann. Appl. Prob. 26, 216–261.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.10.1007/978-1-4612-5320-4CrossRefGoogle Scholar
Djellout, H., Guillin, A. and Wu, L. M. (1999). Large and moderate deviations for estimators of quadratic variational processes of diffusion. Statist. Infer. Stoch. Process. 2, 195–225.10.1023/A:1009950229386CrossRefGoogle Scholar
Fasen, V. (2010). Modeling network traffic by a cluster Poisson input process with heavy and light-tailed file sizes. Queueing Systems 66, 313–350.CrossRefGoogle Scholar
Fasen, V. and Samorodnitsky, G. (2009). A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime. Adv. Appl. Prob. 41, 393–427.CrossRefGoogle Scholar
Faÿ, G., González-Arévalo, B., Mikosch, T. and Samorodnitsky, G. (2006). Modelling teletraffic arrivals by a Poisson cluster process. Queueing Systems 54, 121–140.CrossRefGoogle Scholar
Gao, F. Q. and Zhu, L. (2018). Some asymptotic results for nonlinear Hawkes processes. Stoch. Process. Appl. 128, 4051–4077.CrossRefGoogle Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. G. (2018). Hawkes processes and their applications to finance: a review. Quant. Finance 18, 193–198.CrossRefGoogle Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob 11, 493–503.CrossRefGoogle Scholar
Jessen, A. H., Mikosch, T. and Samorodnitsky, G. (2011). Prediction of outstanding payments in a Poisson cluster model. Scand. Actuarial J. 2011, 214–237.CrossRefGoogle Scholar
Matsui, M. (2014). Prediction in a non-homogeneous Poisson cluster model. Insurance Math. Econom. 55, 10–17.CrossRefGoogle Scholar
Matsui, M. and Mikosch, T. (2010). Prediction in a Poisson cluster model. J. Appl. Prob. 47, 350–366.CrossRefGoogle Scholar
Montgomery-Smith, S. J. (1993). Comparison of sums of independent identically distributed random variables. Prob. Math. Statist. 14, 281–285.Google Scholar
Neyman, J. and Scott, E. L. (1958). Statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 1–43.Google Scholar
Tabassum, H., Hossain, E. and Hossain, J. (2017). Modeling and analysis of uplink non-orthogonal multiple access in large-scale cellular networks using Poisson cluster processes. IEEE Trans. Commun. 65, 3555–3570.Google Scholar
Torrisi, G. L. (2016). Gaussian approximation of nonlinear Hawkes processes. Ann. Appl. Prob. 26, 2106–2140.CrossRefGoogle Scholar
Torrisi, G. L. (2017). Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes. Ann. Inst. H. Poincaré Prob. Statist. 53, 679–700.CrossRefGoogle Scholar
Yi, W., Liu, Y. and Nallanathan, A. (2017). Modeling and analysis of D2D millimeter-wave networks with Poisson cluster processes. IEEE Trans. Commun, Wireless.65, 5574–5588.CrossRefGoogle Scholar
Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Prob. 50, 760–771.CrossRefGoogle Scholar
Zhu, L. (2013). Moderate deviations for Hawkes processes. Statist. Prob. Lett. 83, 885–890.CrossRefGoogle Scholar