Functional large deviations for multivariate regularly varying random walks

Henrik Hult · Filip Lindskog

68 citations0/0 files storedThe Annals of Applied Probability

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Probability and Risk Models Probability and Statistical Research Stochastic processes and statistical mechanics

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.

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References

1997Modelling Extremal Events 1987Extreme Values, Regular Variation and Point Processes 1982Estimates for the probability of ruin with special emphasis on the possibility of large claims 2002Regular variation of GARCH processes 1998Large Deviations of Heavy-Tailed Sums with Applications in Insurance 2004Extremal behavior of regularly varying stochastic processes

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