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Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

Part of: Special processes Mathematical economics Stochastic processes Operations research and management science

Published online by Cambridge University Press:  22 November 2021

Bo Li  and
Guodong Pang
Bo Li*
Affiliation:
Nankai University
Guodong Pang*
Affiliation:
Pennsylvania State University
*
*Postal address: School of Mathematics and LPMC, Nankai University, Tianjin, 300071 China. Email address: libo@nankai.edu.cn
**Postal address: The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, College of Engineering, Pennsylvania State University, University Park, PA 16802, USA. Email address: gup3@psu.edu
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Abstract

We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

Keywords

Shot noise processinfinite-server queuescluster arrival(dependent) random delays of arrivalsdependent noiseslarge-intensity asymptotic regimeGaussian limit processes

MSC classification

Primary: 60G15: Gaussian processes 60G20: Generalized stochastic processes 60G17: Sample path properties
Secondary: 60K25: Queueing theory 90B22: Queues and service 91B30: Risk theory, insurance 60G55: Point processes

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 1190 - 1221
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Basrak, B., Wintenberger, O. and $\check{\text{Z}}$ ugec, P. (2019). On total claim amount for marked Poisson cluster models. Working paper. Available at https://arxiv.org/abs/1903.09387v1.Google Scholar
Biermé, H. and Desolneux, A. (2012). Crossings of smooth shot noise processes. Ann. Appl. Prob. 22, 2240–2281.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures. John Wiley, New York.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I–II, 2nd edn. Springer, New York.Google Scholar
Decreusefond, L. and Nualart, D. (2008). Hitting times for Gaussian processes. Ann. Prob. 36, 319–330.CrossRefGoogle Scholar
Dong, C. and Iksanov, A. (2020). Weak convergence of random processes with immigration at random times. J. Appl. Prob. 57, 250–265.CrossRefGoogle Scholar
Gilchrist, J. H. and Thomas, J. B. (1975). A shot process with burst properties. Adv. Appl. Prob. 7, 527–541.CrossRefGoogle Scholar
Heinrich, L. and Schmidt, V. (1985). Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Prob. 17, 709–730.CrossRefGoogle Scholar
Hsing, T. and Teugels, J. L. (1989). Extremal properties of shot noise processes. Adv. Appl. Prob. 21, 513–525.CrossRefGoogle Scholar
Iglehart, D. L. (1973). Weak convergence of compound stochastic process, I. Stoch. Process. Appl. 1, 11–31.CrossRefGoogle Scholar
Iksanov, A. (2013). Functional limit theorems for renewal shot noise processes with increasing response functions. Stoch. Process. Appl. 123, 1987–2010.CrossRefGoogle Scholar
Iksanov, A. (2016). Renewal Theory for Perturbed Random Walks and Similar Processes. Birkhäuser, Basel.CrossRefGoogle Scholar
Iksanov, A., Marynych, A. and Meiners, M. (2014). Limit theorems for renewal shot noise processes with eventually decreasing response functions. Stoch. Process. Appl. 124, 2132–2170.CrossRefGoogle Scholar
Iksanov, A., Marynych, A. and Meiners, M. (2017). Asymptotics of random processes with immigration I: scaling limits. Bernoulli 23, 1233–1278.Google Scholar
Iksanov, A., Marynych, A. and Meiners, M. (2017). Asymptotics of random processes with immigration II: convergence to stationarity. Bernoulli 23, 1279–1298.Google Scholar
Iksanov, A. and Rashytov, B. (2020). A functional limit theorem for general shot noise processes. J. Appl. Prob. 57, 280–294.CrossRefGoogle Scholar
Klüppelberg, C. and Kühn, C. (2004). Fractional Brownian motion as a weak limit of Poisson shot noise processes—with applications to finance. Stoch. Process. Appl. 113, 333–351.CrossRefGoogle Scholar
Klüppelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125–147.CrossRefGoogle Scholar
Klüppelberg, C., Mikosch, T. and Schärf, A. (2003). Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli 9, 467–496.CrossRefGoogle Scholar
Lane, J. A. (1984). The central limit theorem for the Poisson shot-noise process. J. Appl. Prob. 21, 287–301.CrossRefGoogle Scholar
Lane, J. A. (1987). The Berry–Esseen bound for the Poisson shot-noise. Adv. Appl. Prob. 19, 512–514.CrossRefGoogle Scholar
Marynych, A. (2015). A note on convergence to stationarity of random processes with immigration. Theory Stoch. Process. 20, 84–100.Google Scholar
Marynych, A. and Verovkin, G. (2017). A functional limit theorem for random processes with immigration in the case of heavy tails. Modern Stoch. Theory Appl. 4, 93–108.CrossRefGoogle Scholar
Pang, G. and Taqqu, M. S. (2019). Non-stationary self-similar Gaussian processes as scaling limits of power-law shot noise processes and generalizations of fractional Brownian motion. High Freq. 2, 95–112.CrossRefGoogle Scholar
Pang, G. and Whitt, W. (2012). Infinite-server queues with batch arrivals and dependent service times. Prob. Eng. Inf. Sci. 26, 197–220.CrossRefGoogle Scholar
Pang, G. and Whitt, W. (2013). Two-parameter heavy-traffic limits for infinite-server queues with dependent service times. Queueing Systems 73, 119–146.CrossRefGoogle Scholar
Pang, G. and Zhou, Y. (2018). Functional limit theorems for a new class of non-stationary shot noise processes. Stoch. Process. Appl. 128, 505–544.CrossRefGoogle Scholar
Pang, G. and Zhou, Y. (2018). Two-parameter process limits for infinite-server queues with dependent service times via chaining bounds. Queueing Systems 88, 1–25.CrossRefGoogle Scholar
Pang, G. and Zhou, Y. (2020). Functional limit theorems for shot noise processes with weakly dependent noises. Stoch. Systems 10, 99–123.CrossRefGoogle Scholar
Papoulis, A. (1971). High density shot noise and Gaussianity. J. Appl. Prob. 18, 118–127.CrossRefGoogle Scholar
Ramirez-Perez, F. and Serfling, R. (2001). Shot noise on cluster processes with cluster marks, and studies of long range dependence. Adv. Appl. Prob. 33, 631–651.CrossRefGoogle Scholar
Ramirez-Perez, F. and Serfling, R. (2003). Asymptotic normality of shot noise on Poisson cluster processes with cluster marks. J. Prob. Statist. Sci. 1, 157–172.Google Scholar
Rice, J. (1977). On generalized shot noise. Adv. Appl. Prob. 9, 553–565.CrossRefGoogle Scholar
Samorodnitsky, G. (1995). A class of shot noise models for financial applications. In Proc. Athens Internat. Conf. Appl. Prob. and Time Series, Vol. 1, eds Heyde, C. C., Prohorov, Y. V., Pyke, R. and Rachev, S. T., Springer, Berlin, pp. 332–353.Google Scholar
Shiryaev, A. N. (1996). Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Whitt, W. (1976). Bivariate distributions with given marginals. Ann. Statist. 4, 1280–1289.CrossRefGoogle Scholar
Whitt, W. (1983). Comparing batch delays and customer delays. Bell System Tech J. 62, 2001–2009.CrossRefGoogle Scholar
Whitt, W. (2002). Stochastic-Process Limits: an Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.CrossRefGoogle Scholar