Rhee, C.-H., Blanchet, J., & Zwart, B. (2019). SAMPLE PATH LARGE DEVIATIONS FOR LÉVY PROCESSES AND RANDOM WALKS WITH REGULARLY VARYING INCREMENTS. The Annals of Probability, 47(6), 3551–3605. https://www.jstor.org/stable/26867232

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Abstract

Let X be a Lévy process with regularly varying Lévy measure ν. We obtain sample-path large deviations for scaled processes X̄n(t) and obtain a similar result for random walks with regularly varying increments. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold.

Authors
Chang-Han Rhee, Jose Blanchet, Bert Zwart
Publication date
2019/11/1
Journal
The Annals of Probability
Volume
47
Issue
6
Pages
3551-3605
Publisher
Institute of Mathematical Statistics