## The Annals of Applied Probability

### Functional large deviations for multivariate regularly varying random walks

#### 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

https://projecteuclid.org/euclid.aoap/1133965775

Digital Object Identifier
doi:10.1214/105051605000000502

Mathematical Reviews number (MathSciNet)
MR2187307

Zentralblatt MATH identifier
1166.60309

#### 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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