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Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ken R. Duffy  and
Giovanni Luca Torrisi
Ken R. Duffy*
Affiliation:
National University of Ireland, Maynooth
Giovanni Luca Torrisi*
Affiliation:
Istituto per le Applicazioni del Calcolo
*
∗ Postal address: Hamilton Institute, National University of Ireland, Maynooth, Maynooth Co. Kildare, Ireland. Email address: [email protected]
∗∗ Postal address: Istituto per le Applicazioni del Calcolo ‘Mauro Picone’ (IAC-CNR), Via dei Taurini 19, I-00185 Roma, Italia. Email address: [email protected]
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Abstract

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It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

Keywords

Heavy-tailed distributionsample path large deviationPoisson shot noise

MSC classification

Secondary: 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 688 - 698
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[2] Borovkov, A. A. (2002). On subexponential distributions and the asymptotics of the distribution of the maximum of sequential sums. Sibirsk. Mat. Zh. 43, 1235–1264.Google Scholar
[3] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38), 2nd edn. Springer, New York.Google Scholar
[4] 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
[5] 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.Google Scholar
[6] Gantert, N. (1998). Functional Erdős-Renyi laws for semiexponential random variables. Ann. Prob. 26, 1356–1369.Google Scholar
[7] Klüppelberg, C. and Mikosch, T. (1995). Delay in claim settlement and ruin probability approximations. Scand. Actuarial J. 1995, 154–168.Google Scholar
[8] Konstantopoulos, T. and Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems 28, 215–243.Google Scholar
[9] Lowen, S. B. and Teich, M. C. (1990). Power-law shot noise. IEEE Trans. Inf. Theory 36, 1302–1318.Google Scholar
[10] Rice, S. O. (1944). Mathematical analysis of random noise. Bell System Tech. J. 23, 282–332.Google Scholar
[11] Stabile, G. and Torrisi, G. L. (2010). Large deviations of Poisson shot noise processes under heavy tail semi-exponential conditions. Statist. Prob. Lett. 2010, 1200–1209.Google Scholar