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Functional large deviations for multivariate regularly varying random walks. (English) Zbl 1166.60309
Summary: We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 6 (1969) 17–22 (1970; Zbl 0226.60043), Theor. Probab. Appl. 14, 51–64 (1969); translation from Teor. Veroyatn. Primen. 14, 51–63 (1969; Zbl 0196.21002), Theor. Probab. Appl. 14, 193–208 (1969); translation from Teor. Veroyatn. Primen. 14, 203–216 (1969; Zbl 0196.21003)] 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.
MSC:
60F10Large deviations
60F17Functional limit theorems; invariance principles
60G50Sums of independent random variables; random walks
60B12Limit theorems for vector-valued random variables (infinite-dimensional case)