Hostname: page-component-669899f699-7xsfk Total loading time: 0 Render date: 2025-04-25T14:11:18.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Sample path moderate deviations for non-stationary power-law shot noise processes

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  28 March 2025

Ran Wang ,
Yimin Xiao  and
Qingshan Yang
Ran Wang*
Affiliation:
Wuhan University
Yimin Xiao*
Affiliation:
Michigan State University
Qingshan Yang*
Affiliation:
Northeast Normal University
*
*Postal address: School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China. Email: [email protected]
**Postal address: Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA. Email: [email protected]
***Postal address: KLAS, Key Laboratory of Big Data Analysis of Jilin Province, School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China. Corresponding author email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish a sample path moderate deviation principle for the integrated shot noise process with Poisson arrivals and non-stationary noises. As in Pang and Taqqu (2019), we assume that the noise is conditionally independent given the arrival times, and the distribution of each noise depends on its arrival time. As applications, we derive moderate deviation principles for the workload process and the running maximum process for a stochastic fluid queue with the integrated shot noise process as the input; we also show that a steady-state distribution exists and derive the exact tail asymptotics.

Keywords

Moderate deviationsPoisson shot noisegeneralized fractional Brownian motion

MSC classification

Primary: 60F10: Large deviations
Secondary: 60F17: Functional limit theorems; invariance principles 60G55: Point processes
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 27
Check for updates
Copyright
© The Author(s), 2025. 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.)

Article purchase

Temporarily unavailable

References

Anugu, S. and Pang, G. (2023). Large deviations and long-time behavior of stochastic fluid queues with generalized fractional Brownian motion input. Queueing Systems 105, 47–98.CrossRefGoogle Scholar
Anugu, S. and Pang, G. (2024). Sample path moderate deviations for shot noise processes in the high intensity regime. Stoch. Process. Appl. 176, 104432.CrossRefGoogle Scholar
Armony, M., Israelit, S., Mandelbaum, A., Marmor, Y., Tseytlin, Y. and Yom-Tov, G. (2015). On patient flow in hospitals: A data-based queueing-science perspective. Stoch. Syst. 5, 146–194.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures. John Wiley, New York.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Boxma, O. J., Cahen, E. J., Koops, D. and Mandjes, M. (2019). Linear stochastic fluid networks: Rare-event simulation and Markov modulation. Methodology Comput. Appl. Prob. 21, 125–153.CrossRefGoogle Scholar
Boxma, O. and Mandjes, M. (2021). Shot-noise queueing models. Queueing Systems 99, 121–159.CrossRefGoogle Scholar
Brémaud, P. (2000). An insensitivity property of Lundberg’s estimate for delayed claims. J. Appl. Prob. 37, 914–917.CrossRefGoogle Scholar
Budhiraja, A. and Nyquist, P. (2015). Large deviations for multidimensional state-dependent shot-noise processes. J. Appl. Prob. 52, 1097–1114.CrossRefGoogle Scholar
Chang, C. S., Yao, D. and Zajic, T. (1999). Large deviations, moderate deviations, and queues with long-range dependent input. Adv. Appl. Prob. 31, 254–277.CrossRefGoogle Scholar
Chang, C. S., Yao, D. and Zajic, T. (2003). Large deviations, long-range dependence, and queues. In Stochastic Modeling and Optimization, eds D. D. Yao, X. Y. Zhou and H. Zhang. Springer, New York, pp. 245–277.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications. Springer, New York.CrossRefGoogle Scholar
Duffield, N. and O’Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Camb. Philos. Soc. 118, 363–374.CrossRefGoogle Scholar
Ganesh, A., Macci, C. and Torrisi, G. (2005). Sample path large deviations principles for Poisson shot noise processes, and applications. Electron. J. Prob. 10, 1026–1043.CrossRefGoogle Scholar
Ganesh, A., O’Connell, N. and Wischik, D. (2004). Big Queues (Lect. Notes Math. 1838). Springer, Berlin.CrossRefGoogle Scholar
Ganesh, A. and Torrisi, G. L. (2006). A class of risk processes with delayed claims: Ruin probability estimates under heavy tail conditions. J. Appl. Prob. 43, 916–926.CrossRefGoogle Scholar
Gao, F. (1997). A note on Cramér’s theorem. In Séminaire de Probabilités XXXI (Lect. Notes Math. 1655), eds J. Azéma, M. Yor and M. Emery. Springer, Berlin, pp. 77–79.Google Scholar
Gao, F. and Wang, Y. (2020). Functional central limit theorems and moderate deviations for Poisson cluster processes. Adv. Appl. Prob. 52, 916–941.CrossRefGoogle Scholar
Garsia, A., Rodemich, E. and Rumsey, H. Jr. (1970). A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20, 565–578.CrossRefGoogle Scholar
Ichiba, T., Pang, G. D. and Taqqu, M. S. (2022). Path properties of a generalized fractional Brownian motion. J. Theoret. Prob. 35, 550–574.CrossRefGoogle Scholar
Ichiba, T., Pang, G. D. and Taqqu, M. S. (2024). Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in finance. To appear in Finance Stoch. Google Scholar
Iksanov, A. (2016). Renewal Theory for Perturbed Random Walks and Similar Processes. Birkhäuser/Springer, Cham.CrossRefGoogle Scholar
Jiang, H. and Yang, Q. S. (2021). Sample path large deviations for the multiplicative Poisson shot noise process with compensation. Stochastics 93, 447–477.CrossRefGoogle Scholar
Khoshnevisan, D. (2014). Analysis of Stochastic Partial Differential Equations (CBMS Regional Conference Series in Mathematics 119). American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Kingman, J. (1993). Poisson Processes. Oxford Science Publications, Oxford.Google 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
Kozachenko, Y., Vasylyk, O. and Sottinen, T. (2002). Path space large deviations of a large buffer with Gaussian input traffic. Queueing Systems 42, 113–129.CrossRefGoogle Scholar
Lane, J. A. (1984). The central limit theorem for the Poisson shot-noise process. J. Appl. Prob. 21, 287–301.CrossRefGoogle Scholar
Lowen, S. B. and Teich, M. C. (2005). Fractal-Based Point Processes. John Wiley, Chichester.CrossRefGoogle Scholar
Macci, C. and Pacchiarotti, B. (2017). Exponential tightness for Gaussian processes, with applications to some sequences of weighted means. Stochastics 89, 469–484.CrossRefGoogle Scholar
Majewski, K. (2003). Large deviations for multi-dimensional reflected fractional Brownian motion. Stoch. Stoch. Reports 75, 233–257. (Corrigendum: Stoch. Stoch. Reports 76 (2004), 479.)CrossRefGoogle Scholar
Majewski, K. (2005). Sample path moderate deviations for a family of long-range dependent traffic and associated queue length processes. Stochastics 77, 81–107.CrossRefGoogle Scholar
Pang, G. D. and Taqqu, M. S. (2019). Nonstationary 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. D. and Zhou, Y. (2017). Two-parameter process limits for an infinite-server queue with arrival dependent service times. Stoch. Process. Appl. 127, 1375–1416.CrossRefGoogle Scholar
Pang, G. D. 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. D. and Zhou, Y. (2020). Functional limit theorems for shot noise processes with weakly dependent noises. Stoch. Syst. 10, 99–123.CrossRefGoogle Scholar
Pipiras, V. and Taqqu, M. S. (2017). Long-Range Dependence and Self-Similarity. Cambridge University Press.CrossRefGoogle Scholar
Prabhu, N. U. (1998). Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication. Springer, Berlin.CrossRefGoogle Scholar
Puhalskii, A. (2001). Large Deviations and Idempotent Probability. Chapman and Hall/CRC, New York.CrossRefGoogle Scholar
Rajput, B. and Rosinski, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.CrossRefGoogle Scholar
Rhee, C. H., Blanchet, J. and Zwart, B. (2019). Sample path large deviations for Lévy processes and random walks with regularly varying increments. Ann. Prob. 47, 3551–3605.CrossRefGoogle Scholar
Schied, A. (1995). Criteria for exponential tightness in path spaces. Discussion paper SFB 303, Bonn.Google Scholar
Schied, A. (1997). Moderate deviations and functional LIL for super-Brownian motion. Stoch. Process. Appl. 72, 11–25.CrossRefGoogle Scholar
Schmidt, T. (2017). Shot-noise processes in finance. In From Statistics to Mathematical Finance, eds D. Ferger, W. González Manteiga, T. Schmidt and J.-L. Wang. Springer, Cham, pp. 367–385.CrossRefGoogle Scholar
Van Der Ziel, A. (1979). Flicker noise in electronic devices. Adv. Electron. Electron Phys. 49, 225–297.CrossRefGoogle Scholar
Walsh, J. (1986). An Introduction to Stochastic Partial Differential Equations (Lect. Notes Math. 1180). Springer, Berlin.CrossRefGoogle Scholar
Wang, R. and Xiao, Y. (2022). Exact uniform modulus of continuity and Chung’s LIL for the generalized fractional Brownian motion. J. Theoret. Prob. 35, 2442–2479.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, Berlin.CrossRefGoogle Scholar
Yan, J. and Gao, F. (2012). Exponential martingale and large deviations for a Cox risk process with Poisson shot noise intensity. J. Math. Anal. Appl. 394, 74–83.CrossRefGoogle Scholar