Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-07-27T14:01:36.649Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail asymptotics and precise large deviations for some Poisson cluster processes

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  26 July 2024

Fabien Baeriswyl Open the ORCID record for Fabien Baeriswyl [Opens in a new window] ,
Valérie Chavez-Demoulin  and
Olivier Wintenberger
Fabien Baeriswyl*
Affiliation:
Département des Opérations, Université de Lausanne and Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université
Valérie Chavez-Demoulin*
Affiliation:
Département des Opérations, Université de Lausanne
Olivier Wintenberger*
Affiliation:
Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université
*
*Postal address: Département des Opérations, Anthropole, CH-1015 Lausanne, Suisse. Emails: fabien.baeriswyl@unil.ch and fabien.baeriswyl@sorbonne-universite.fr
**Postal address: Département des Opérations, Anthropole, CH-1015 Lausanne, Suisse.
***Postal address: Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, Campus Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [6, 19], notably relying on Karamata’s Tauberian Theorem to do so. We use these asymptotics to derive precise large-deviation results in the fashion of [32] for the just-mentioned processes.

Keywords

Renewal Poisson cluster processHawkes processrandom maximarandom sums

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G55: Point processes 60F10: Large deviations
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 37
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. and Foss, S. (2018). Regular variation in a fixed-point problem for single- and multiclass branching processes and queues. Advances in Applied Probability. 50, 47–61.CrossRefGoogle Scholar
Basrak, B., Conroy, M., Olvera-Cravioto, M. and Palmowski, Z. (2022). Importance sampling for maxima on trees. Stochastic Processes and Their Applications. 148, 139–179.CrossRefGoogle Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Annals of Applied Probability. 908–920.Google Scholar
Basrak, B., Kulik, R. and Palmowski, Z. (2013). Heavy-tailed branching process with immigration. Stochastic Models. 29, 413–434.CrossRefGoogle Scholar
Basrak, B., Milinc̆ević, N. and Žugec, P. (2023). On extremes of random clusters and marked renewal cluster processes. Journal of Applied Probability. 60, 367–381.CrossRefGoogle Scholar
Basrak, B., Wintenberger, O. and Žugec, P. (2019). On the total claim amount for marked Poisson cluster models. Advances in Applied Probability. 51, 541–569.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation, Vol. 27. Cambridge University Press.Google Scholar
Brémaud., P. (2020). Point Process Calculus in Time and Space: An Introduction with Applications, Vol. 98. Springer Nature.CrossRefGoogle Scholar
Chavez-Demoulin, V., Davison, A. C. and McNeil, A. J. (2005). Estimating value-at-risk: a point process approach. Quantitative Finance. 5, 227–234.CrossRefGoogle Scholar
Chen, N., Litvak, N. and Olvera-Cravioto, M. Pagerank in scale-free random graphs. In Algorithms and Models for the Web Graph: 11th International Workshop, WAW 2014, Beijing, China, December 17–18, 2014, Proceedings 11, pages 120–131. SpringerCrossRefGoogle Scholar
Chen, N., Litvak, N. and Olvera-Cravioto, M. (2017). Generalized pagerank on directed configuration networks. Random Structures & Algorithms. 51, 237–274.CrossRefGoogle Scholar
Cline, D. B. H. and Hsing, T. (1991). Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Preprint, Texas A&M University, 501.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, Probability and Its Applications. Springer-Verlag, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Volume II, General Theory and Structure. Springer, New York.CrossRefGoogle Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction, Vol. 21. Springer.CrossRefGoogle Scholar
Denisov, D., Foss, S. and Korshunov, D. (2010). Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli, 971–994.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, K. and Mikosch, T. (2013). Modelling Extremal Events: for Insurance and Finance, Vol. 33. Springer Science & Business Media.Google Scholar
Ernst, P. A., Asmussen, S. and Hasenbein, J. J. (2018). Stability and busy periods in a multiclass queue with state-dependent arrival rates. Queueing Systems. 90, 207–224.CrossRefGoogle Scholar
Faÿ, G., González-Arévalo, B., Mikosch, T. and Samorodnitsky, G. (2006). Modeling teletraffic arrivals by a poisson cluster process. Queueing Systems. 54, 121–140.CrossRefGoogle Scholar
Foufoula-Georgiou, E. and Lettenmaier, D. P. (1987). A Markov renewal model for rainfall occurrences. Water resources research. 23, 875–884.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. Quantitative Finance. 18, 193–198.CrossRefGoogle Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. Journal of Applied Probability. 11, 493–503.CrossRefGoogle Scholar
Heyde, C. C. (1967). On large deviation problems for sums of random variables which are not attracted to the normal law. The Annals of Mathematical Statistics. 38, 1575–1578.CrossRefGoogle Scholar
Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publications de l’Institut Mathématique. 80, 121–140.CrossRefGoogle Scholar
Hult, H. and Samorodnitsky, G. (2008). Tail probabilities for infinite series of regularly varying random vectors. Bernoulli. 14, 838–864.CrossRefGoogle Scholar
Jelenković, P. R. and Olvera-Cravioto, M. (2010). Information ranking and power laws on trees. Advances in Applied Probability. 42, 1057–1093.CrossRefGoogle Scholar
Jessen, H. A. and Mikosch, T. (2006). Regularly varying functions. Publications de l’Institut Mathématique. 80, 171–192.CrossRefGoogle Scholar
Karabash, D. and Zhu, L. (2015). Limit theorems for marked Hawkes processes with application to a risk model. Stochastic Models. 31, 433–451.CrossRefGoogle Scholar
Karamata, J. (1933). Sur un mode de croissance régulière: théorèmes fondamentaux. Bulletin de la Société Mathématique de France. 61, 55–62.CrossRefGoogle Scholar
Karpelevich, F., Kelbert, M. Y. and Suhov, Y. M. (1994). Higher-order Lindley equations. Stochastic Processes and their Applications. 53, 65–96.CrossRefGoogle Scholar
Klüppelberg, C. and Mikosch, T. (1997). Large deviations of heavy-tailed random sums with applications in insurance and finance. Journal of Applied Probability. 34, 293–308.CrossRefGoogle Scholar
Lindskog, F., Resnick, S. I. and Roy, J. (2014). Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. Probability Surveys. 11, 270–314.CrossRefGoogle Scholar
Litvak, N., Scheinhardt, W. R. W. and Volkovich, Y. (2007). In-degree and Pagerank: why do they follow similar power laws? Internet Mathematics. 4, 175–198.CrossRefGoogle Scholar
Liu, L. (2009). Precise large deviations for dependent random variables with heavy tails. Statistics & Probability Letters. 79, 1290–1298.CrossRefGoogle Scholar
Markovich, N. (2023). Extremal properties of evolving networks: local dependence and heavy tails. Annals of Operations Research. 1–32. https://doi.org/10.1007/s10479-023-05175-y Google Scholar
Markovich, N. M. and Rodionov, I. V. (2020). Maxima and sums of non-stationary random length sequences. Extremes. 23, 451–464.CrossRefGoogle Scholar
Mikosch, T. (2009). Non-life insurance mathematics: an introduction with the Poisson process. Springer Science & Business Media.CrossRefGoogle Scholar
Mikosch, T. and Nagaev, A. V. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes. 1, 81–110.CrossRefGoogle Scholar
Mikosch, T. and Wintenberger, O. (2024). Extreme Value Theory for Time Series: Models with Power-Law Tails. Springer Series in Operations Research and Financial Engineering. Springer Cham. https://link.springer.com/book/9783031591556 Google Scholar
Musmeci, F. and Vere-Jones, D. (1992). A space-time clustering model for historical earthquakes. Annals of the Institute of Statistical Mathematics. 44, 1–11.CrossRefGoogle Scholar
Nagaev, A. V. (1969). Integral limit theorems taking large deviations into account when Cramér’s condition does not hold. I. Theory of Probability & Its Applications. 14, 51–64.CrossRefGoogle Scholar
Nagaev, A. V. (1969). Integral limit theorems taking large deviations into account when Cramér’s condition does not hold. II. Theory of Probability & Its Applications. 14, 193–208.CrossRefGoogle Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. The Annals of Probability. 7, 745–789.CrossRefGoogle Scholar
Ng, K. W., Tang, Q., Yan, J.-A. and Yang, H. (2004). Precise large deviations for sums of random variables with consistently varying tails. Journal of Applied Probability. 41, 93–107.CrossRefGoogle Scholar
Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association. 83, 9–27.CrossRefGoogle Scholar
Olvera-Cravioto, M. (2021). PageRank’s behavior under degree correlations. The Annals of Applied Probability. 31, 1403–1442.CrossRefGoogle Scholar
Onof, C. et al. (2000). Rainfall modelling using Poisson-cluster processes: a review of developments. Stochastic Environmental Research and Risk Assessment. 14, 384–411.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extremes Values, Regular Variation and Point Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Science & Business Media.Google Scholar
Reynaud-Bouret, P. and Schbath, S. (2010). Adaptive estimation for Hawkes processes: application to genome analysis. The Annals of Statistics. 38, 2781–2822.CrossRefGoogle Scholar
Robert, C. Y and Segers, J. (2008). Tails of random sums of a heavy-tailed number of light-tailed terms. Insurance: Mathematics and Economics. 43, 85–92.Google Scholar
Segers, J., Zhao, Y. and Meinguet, T. (2016). Polar decomposition of regularly varying time series in star-shaped metric spaces. Preprint, arXiv:1604.00241.Google Scholar
Stabile, G. and Torrisi, G. L. (2010). Risk processes with non-stationary Hawkes claims arrivals. Methodology and Computing in Applied Probability. 12, 415–429.CrossRefGoogle Scholar
Swishchuk, A. (2018). Risk model based on compound Hawkes process. Wilmott. 94, 50–57.CrossRefGoogle Scholar
Tang, Q. (2006). Insensitivity to Negative Dependence of the Asymptotic Behavior of Precise Large Deviations. Electronic Journal of Probability. 11, 107–120.CrossRefGoogle Scholar
Tang, Q., Su, C., Jiang, T. and Zhang, J. (2001). Large deviations for heavy-tailed random sums in compound renewal model. Statistics & Probability Letters. 52, 91–100.CrossRefGoogle Scholar
Tillier, C. and Wintenberger, O. (2018). Regular variation of a random length sequence of random variables and application to risk assessment. Extremes. 21, 27–56.CrossRefGoogle Scholar
Vere-Jones, D. and Ozaki, T. (1982). Some examples of statistical estimation applied to earthquake data: I. Cyclic Poisson and self-exciting models. Annals of the Institute of Statistical Mathematics. 34, 189–207.CrossRefGoogle Scholar
Volkovich, Y. and Litvak, N. Asymptotic analysis for personalized web search. Advances in Applied Probability. 42, 577–604.CrossRefGoogle Scholar