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.