Functional large deviations for multivariate regularly varying random walks

By Henrik Hult, Filip Lindskog, Thomas Mikosch, Gennady Samorodnitsky. 2005

109 citations recorded by the source corpus.

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 cAdlAg 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 se-ments. 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.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/dbc5d7d80ef9103b23eb6d4caa1f74048080b5df

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Citations in the corpus

State-dependent importance sampling for regularly varying random walks (2008, background): “Recently, Hult and Lindskog (2005) and Hult et al. (2005) have developed sample path large deviations for regularly varying processes and other results that parallel some of the development reached by the light-tailed counterpart.”
Regularly Varying Measures on Metric Spaces: Hidden Regular Variation and Hidden Jumps (2013, unknown): “See, for instance, [8, 16, 17, 19, 20, 24, 26, 37, 40, 42, 46].”
Asymptotics for the ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims (2011, background): “[16], who mainly focused on multivariate regularly varying random walks and provided sharp asymptotics for general ruin boundaries.”
Measures of serial extremal dependence and their estimation (2013, background): “[33], Mikosch and Wintenberger [49, 50], and to random sets in Mikosch et al.”
REGULAR VARIATION FOR MEASURES ON METRIC SPACES (2006, unknown): “1 with cn = [nP (|Z1| > n)]−1, see [14], Theorem 2.”
Asymptotics for a bidimensional risk model with two geometric Lévy price processes (2019, background): “Ruin for multi-dimensional heavy-tailed processes was initially studied by [7], who mainly focused on multivariate regularly varying random walks and provided sharp asymptotics for general ruin boundaries.”