Sample-path large deviations for a class of heavy-tailed Markov additive processes

TL;DR

This paper establishes a sample-path large deviations principle for heavy-tailed Markov additive processes driven by affine recursions, revealing that most unlikely paths are step functions with jumps, under certain heavy-tailed conditions.

Contribution

It introduces a large deviations framework for additive processes with heavy tails, extending previous results to include signed increments and the $M_1'$ topology.

Findings

Most likely paths are step functions with jumps.

Heavy-tailed distributions arise under Kesten's condition.

Large deviations are characterized in the $M_1'$ topology.

Abstract

For a class of additive processes driven by the affine recursion $X_{n+1} = A_n X_n + B_n$, we develop a sample-path large deviations principle in the $M_1'$ topology on $D [0,1]$. We allow $B_n$ to have both signs and focus on the case where Kesten's condition holds on $A_1$, leading to heavy-tailed distributions. The most likely paths in our large deviations results are step functions with both positive and negative jumps.