## The Annals of Probability

### Sample path large deviations for Lévy processes and random walks with regularly varying increments

#### 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 6 (2019), 3551-3605.

Dates
Revised: October 2018
First available in Project Euclid: 2 December 2019

https://projecteuclid.org/euclid.aop/1575277337

Digital Object Identifier
doi:10.1214/18-AOP1319

Mathematical Reviews number (MathSciNet)
MR4038038

Subjects
Primary: 60F10: Large deviations 60G17: Sample path properties
Secondary: 60B10: Convergence of probability measures

#### Citation

Rhee, Chang-Han; Blanchet, Jose; Zwart, Bert. Sample path large deviations for Lévy processes and random walks with regularly varying increments. Ann. Probab. 47 (2019), no. 6, 3551--3605. doi:10.1214/18-AOP1319. https://projecteuclid.org/euclid.aop/1575277337

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