The Annals of Applied Probability

Functional large deviations for multivariate regularly varying random walks

Henrik Hult, Filip Lindskog, Thomas Mikosch, and Gennady Samorodnitsky

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Abstract

We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser. Fiz.–Mat. Nauk 6 (1969) 17–22, Theory Probab. Appl. 14 (1969) 51–64, 193–208] on large deviations for sums of i.i.d. regularly varying random variables to partial sum processes of i.i.d. regularly varying vectors. The results are stated in terms of a heavy-tailed large deviation principle on the space of càdlàg functions. We illustrate how these results can be applied to functionals of the partial sum process, including ruin probabilities for multivariate random walks and long strange segments. These results make precise the idea of heavy-tailed large deviation heuristics: in an asymptotic sense, only the largest step contributes to the extremal behavior of a multivariate random walk.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 4 (2005), 2651-2680.

Dates
First available in Project Euclid: 7 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1133965775

Digital Object Identifier
doi:10.1214/105051605000000502

Mathematical Reviews number (MathSciNet)
MR2187307

Zentralblatt MATH identifier
1166.60309

Subjects
Primary: 60F10: Large deviations 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
Large deviations regular variation functional limit theorems random walks

Citation

Hult, Henrik; Lindskog, Filip; Mikosch, Thomas; Samorodnitsky, Gennady. Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Probab. 15 (2005), no. 4, 2651--2680. doi:10.1214/105051605000000502. https://projecteuclid.org/euclid.aoap/1133965775


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References

  • Araujo, A. and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.
  • Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95--115.
  • Billingsley, P. (1968). Convergence of Probability Measures, 1st ed. Wiley, New York.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Cline, D. B. H. and Hsing, T. (1998). Large deviation probabilities for sums of random variables with heavy or subexponential tails. Technical report, Texas A&M Univ.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  • Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55--72.
  • Gihman, I. I. and Skorohod, A. V. (1975). The Theory of Stochastic Processes II. Springer, Berlin.
  • Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126--166.
  • de Haan, L. and Lin, T. (2001). On convergence toward an extreme value limit in $C[0,1]$. Ann. Probab. 29 467--483.
  • Hult, H. and Lindskog, F. (2005). Extremal behavior for regularly varying stochastic processes. Stochastic Process. Appl. 115 249--274.
  • Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie-Verlag, Berlin.
  • Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207--248.
  • Lindskog, F. (2004). Multivariate extremes and regular variation for stochastic processes. Ph.D. thesis, Dept. Mathematics, Swiss Federal Institute of Technology, Switzerland. Available at e-collection.ethbib.ethz.ch/cgi-bin/show.pl?type=diss&nr=15319.
  • Mansfield, P., Rachev, S. and Samorodnitsky, G. (2001). Long strange segments of a stochastic process and long range dependence. Ann. Appl. Probab. 11 878--921.
  • Mikosch, T. and Nagaev, A.V. (1998). Large deviations of heavy-tailed sums with applications to insurance. Extremes 1 81--110.
  • Nagaev, A. V. (1969). Limit theorems for large deviations where Cramér's conditions are violated. Izv. Akad. Nauk UzSSR Ser. Fiz.--Mat. Nauk 13 17--22. (In Russian.)
  • Nagaev, A. V. (1969). Integral limit theorems for large deviations when Cramér's condition is not fulfilled I, II. Theory Probab. Appl. 14 51--64, 193--208.
  • Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745--789.
  • Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford Univ. Press.
  • Rachev, S. and Samorodnitsky, G. (2001). Long strange segments in a long range dependent moving average. Stochastic Process. Appl. 93 119--148.
  • Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66--138.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Rvačeva, E. L. (1962). On domains of attraction of multi-dimensional distributions. In Select. Transl. Math. Statist. Probab. 2 183--205. Amer. Math. Soc., Providence, RI.