Sample Path Large Deviations for L\'evy Processes and Random Walks with Regularly Varying Increments

TL;DR

This paper establishes sample-path large deviations for Lévy processes and random walks with regularly varying increments, providing detailed asymptotics for rare events involving multiple large jumps and exploring connections with classical large deviations theory.

Contribution

It extends large deviation results to processes with regularly varying jumps, highlighting scenarios with multiple big jumps and clarifying the limitations of classical principles.

Findings

Sample-path large deviations for scaled Lévy processes and random walks.

Asymptotic estimates for events requiring multiple large jumps.

Weak large deviation principle with logarithmic speed, but no full LDP.

Abstract

Let $X$ be a L\'evy process with regularly varying L\'evy 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. 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.