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November 2019 Sample path large deviations for Lévy processes and random walks with regularly varying increments
Chang-Han Rhee, Jose Blanchet, Bert Zwart
Ann. Probab. 47(6): 3551-3605 (November 2019). DOI: 10.1214/18-AOP1319

Abstract

Let $X$ be a Lévy process with regularly varying Lévy measure $\nu$. We obtain sample-path large deviations for scaled processes $\bar{X}_{n}(t)\triangleq X(nt)/n$ 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.

Citation

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Chang-Han Rhee. Jose Blanchet. Bert Zwart. "Sample path large deviations for Lévy processes and random walks with regularly varying increments." Ann. Probab. 47 (6) 3551 - 3605, November 2019. https://doi.org/10.1214/18-AOP1319

Information

Received: 1 September 2016; Revised: 1 October 2018; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212167
MathSciNet: MR4038038
Digital Object Identifier: 10.1214/18-AOP1319

Subjects:
Primary: 60F10, 60G17
Secondary: 60B10

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • November 2019
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