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Regular variation of a random length sequence of random variables and application to risk assessment

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Abstract

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence C(N) = (C 1,…,C N ) of random length N, where C(N) comes from the product of a matrix A(N) of random size N × N and a random sequence X(N) of random length N. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the extremal behavior of ∥C(N)∥, for some norm ∥⋅∥. We propose a generalization of Breiman’s Lemma that gives way to a tail estimate of ∥C(N)∥ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our main result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.

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References

  • Asmussen, S.: Applied Probabilities and Queues. Springer-Verlag, New York (2003)

    MATH  Google Scholar 

  • Asmussen, S., Albrecher, H.: Ruin probabilities. World Scientific Publishing C0. Pte. Ltd. 2nd edn. p. 602 (2010)

  • Barbe, P., Fougères, A.L., Genest, C.: On the tail behavior of sums of dependent risks. Astin Bulletin. 36(2), 361–373 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Basrak, B., Davis, R., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12(3), 908–920 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Basrak, B., Davis, R., Mikosch, T.: Regular variation of GARCH processes. Stochastic Processes and their Applications. 99(1), 95–115 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Basrak, B., Segers, J.: Regularly varying multivariate time series. Stochastic Processes and their Applications. 119(4), 1055–1080 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Bertail, P., Clemencon, S., Tressou, J.: A storage model with random release rate for modelling exposure to food contaminants. Math. Biosci. Eng. 5, 35–60 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Bertail, P., Clemencon, S., Tressou, J.: Statistical analysis of a dynamic model for food contaminant exposure with applications to dietary methylmercury contamination. J. Biol. Dyn. 4, 212–234 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation, p. 494. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

  • Billingsley, P.: Convergence of probability measures. 2nd edition. Wiley series in probability and statistics (1999)

  • Bougerol, P.: Kalman filtering with random coefficients and contractions. SIAM J. Control Optim. archive 31(4), 942–959 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Cline, D.B.H., Samorodnitsky, G.: Subexponentiality of the product of independent random variables. Stochastic Processes and their Applications 49, 75–98 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Drees, H., Janssen, A.: A stochastic volatility model with flexible extremal dependence structure. Bernoulli 22, 1448–1490 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Embrechts, P., Kluppelberg, C., Mikosch, T.: Modelling extremal events for insurance and finance, Applications of Mathematics. Springer-Verlag (1997)

  • FAO/WHO: Evaluation of certain food additives and contaminants for methylmercury. Sixty first report of the Joint FAO/WHO Expert Committee on Food Additives, Technical Report Series (2003)

  • Gibaldi, M., Perrier, D.: Pharmacokinetics. Drugs and the pharmaceutical sciences: a series of textbooks and monographs, 2nd edn. Marcel Dekker, New York (1982)

    Google Scholar 

  • Kyprianou, A.E.: Gerber Shiu Risk Theory. EAA Series. Springer International Publishing (2013)

  • Gudendorf, G., Segers, J.: 6: Copula Theory and its Applications, Lecture Notes in Statistics 198, pp. 127–145. Springer-Verlag, Berlin (2010)

  • Grandell, J.: Aspects of risk theory Springer Series in Statistics: Probability and its Applications. Springer-Verlag, New York (1991)

    Book  MATH  Google Scholar 

  • Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publications de l’institut mathematiqué, Nouvelle sé,rie 80(94), 121–140 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Jessen, A.H., Mikosch, T.: Regularly varying functions. Publications de l’institut mathematiqué. Nouvelle serie, tome 80(94), 171–192 (2006). doi:10.2298/PIM0694171H

    MATH  Google Scholar 

  • Galambos, J., Simonelli, I.: Bonferroni-Type Inequalities with Applications. Probability and Its Applications, p. 269. Springer-Verlag, New York (1996)

    MATH  Google Scholar 

  • Li, X., Wu, J.: Asymptotic tail behavior of Poisson shot-noise processes with interdependence between shock and arrival time. Statistics and Probability Letters 88, 15–26 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Lindskog, F., Resnick, S.I., Roy, J.: Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. Probability Surv. 11, 270–314 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Mikosch, T.: Regular Variation, Subexponentiality and their Applications in Probability Theorym. Eindhoven University of Technology, vol. 99, no. 13 of EURANDOM report, ISSN 1389-2355 (1999)

  • Mikosch, T.: Non life Insurance Mathematics, 2nd edn. Springer-Verlag, Berlin (2010)

    MATH  Google Scholar 

  • Olvera-Cravioto, M.: Asymptotics for weighted random sums. Adv. Appl. Prob. 44, 1142–1172 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Perfekt, R.: Extremal behaviour of stationary markov chains with applications. Ann. Appl. Probab. 4(2), 529–548 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Resnick, S.I.: Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5(4), 303–336 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Resnick, S.I.: On the foundations of multivariate Heavy-Tail analysis. J. Appl. Probab. 41, 191–212 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Resnick, S.I.: Heavy-tail Phenomena and Statistical Modeling Springer Series in Operations Research and Financial Engineering. Springer-Verlag, New-York (2007)

    Google Scholar 

  • Leadbetter, M.R., Rootzen, H.: Extremal theory for stochastic processes. Ann. Probab. 16(2), 431–478 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Schmidt, T.: Catastrophe Insurance modeled by Shot-Noise Processes. Risks 2, 3–24 (2014)

    Article  Google Scholar 

  • Weng, C., Zhang, Y., Tang, K.S.: Tail behavior of Poisson Shot Noise Processes under heavy-tailed shocks and Actuarial applications. Methodol. Comput. Appl. Probab. 15(3), 665–682 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

Charles Tillier would like to thank the Institute of Mathematical Sciences of the University of Copenhagen for hospitality and financial support when visiting Olivier Wintenberger. Financial supports by the ANR network AMERISKA ANR 14 CE20 0006 01 are also gratefully acknowledged by both authors. We would like to thank the referees and the Associate Editor for insightful comments which led to an improvement of the paper.

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Authors and Affiliations

  1. Université Paris Nanterre, 200 avenue de la république, 92000, Nanterre, France

    Charles Tillier

  2. Université Pierre et Marie Curie, LSTA, 4 place Jussieu, 75005, Paris, France

    Charles Tillier & Olivier Wintenberger

  3. Department of Mathematical Sciences, UCPH, Universitetsparken 5, 2100, Copenhague, Danemark

    Olivier Wintenberger

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  1. Charles Tillier
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  2. Olivier Wintenberger
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Correspondence to Charles Tillier.

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Tillier, C., Wintenberger, O. Regular variation of a random length sequence of random variables and application to risk assessment. Extremes 21, 27–56 (2018). https://doi.org/10.1007/s10687-017-0297-1

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  • DOI: https://doi.org/10.1007/s10687-017-0297-1

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