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

First available in Project Euclid: 7 December 2005

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Zentralblatt MATH identifier

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)

Large deviations regular variation functional limit theorems random walks


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.

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