Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T13:16:46.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations for multidimensional state-dependent shot-noise processes

Part of: Special processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 March 2016

Amarjit Budhiraja  and
Pierre Nyquist
Amarjit Budhiraja*
Affiliation:
University of North Carolina at Chapel Hill
Pierre Nyquist*
Affiliation:
Brown University
*
∗Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA. Email address: [email protected]
∗∗Postal address: Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Shot-noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory, and in the engineering sciences. In this paper we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot-noise processes. The result covers previously known large deviation results for one-dimensional state-independent shot-noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate convergence of certain controlled versions of the original processes together with relevant results on existence and uniqueness.

Keywords

Large deviationsPoisson shot-noisePoisson random measurevariational representations

MSC classification

Primary: 60F10: Large deviations 60G55: Point processes
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1097 - 1114
Copyright
Copyright © Applied Probability Trust 2015 

References

[1] Boue, M. and Dupuis, P. (1998). A variational representation for certain functionals of Brownian motion. Ann. Prob. 26, 1641–1659.CrossRefGoogle Scholar
[2] Brockwell, P. J., Resnick, S. I. and Tweedie, R. L. (1982). Storage processes with general release rule and additive inputs. Adv. Appl. Prob. 14, 392–433.Google Scholar
[3] Budhiraja, A., Chen, J. and Dupuis, P. (2013). Large deviations for stochastic partial differential equations driven by a Poisson random measure. Stoch. Process. Appl. 123, 523–560.Google Scholar
[4] Budhiraja, A., Dupuis, P. and Maroulas, V. (2011). Variational representations for continuous time processes. Ann. Inst. H. Poincaré Prob. Statist. 47, 725–747.Google Scholar
[5] Daley, D. J. (1971). The definition of a multi-dimensional generalization of shot noise. J. Appl. Prob. 8, 128–135.Google Scholar
[6] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.Google Scholar
[7] Doney, R. A. and O'Brien, G. L. (1991). Loud shot noise. Ann. Appl. Prob. 1, 88–103.Google Scholar
[8] Ganesh, A., Macci, C. and Torrisi, G. L. (2005). Sample path large deviations principles for Poisson shot noise processes, and applications. Electron. J. Prob. 10, 1026–1043.Google Scholar
[9] Huffer, F. W. (1987). Inequalities for the M/G/8 queue and related shot noise processes. J. Appl. Prob. 24, 978–989.Google Scholar
[10] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes , 2nd edn. Springer, Berlin.Google Scholar
[11] KlüPpelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125–147.CrossRefGoogle Scholar
[12] KlüPpelberg, C., Mikosch, T. and SchäRf, A. (2003). Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli 9, 467–496.Google Scholar
[13] Lane, J. A. (1984). The central limit theorem for the Poisson shot-noise process. J. Appl. Prob. 21, 287–301.Google Scholar
[14] Lund, R. B. (1996). The stability of storage models with shot noise input. J. Appl. Prob. 33, 830–839.Google Scholar
[15] Lund, R., Mccormick, W. P. and Xiao, Y. (2004). Limiting properties of Poisson shot noise processes. J. Appl. Prob. 41, 911–918.CrossRefGoogle Scholar
[16] Mccormick, W. P. (1997). Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Prob. 34, 643–656.Google Scholar
[17] Rice, J. (1977). On generalized shot noise. Adv. Appl. Prob. 9, 553–565.Google Scholar
[18] Rice, S. O. (1945). Mathematical analysis of random noise. Bell System Tech. J. 24, 46–156.CrossRefGoogle Scholar
[19] Samorodnitsky, G. (1998). Tail behavior of some shot noise processes. In A Practical Guide to Heavy Tails , Birkhäuser, Boston, MA, pp. 473–478.Google Scholar
[20] Torrisi, G. L. (2004). Simulating the ruin probability of risk processes with delay in claim settlement. Stoch. Process. Appl. 112, 225–244.Google Scholar