Fluctuation theory for Markov random walks

TL;DR

This paper extends fundamental fluctuation theorems for real-valued random walks to the Markov-modulated case, analyzing how excursions influence divergence types and moments of passage times.

Contribution

It introduces a novel approach using excursion measures to generalize fluctuation theorems for Markov random walks beyond iid assumptions.

Findings

Generalization of fluctuation theorems to Markov-modulated walks

Identification of excursion measures affecting moment existence

Counterexamples illustrating non-trivial extensions

Abstract

Two fundamental theorems by Spitzer/Erickson and Kesten/Maller on the fluctuation type (positive divergence, negative divergence or oscillation) of a real-valued random walk $(S_{n})_{n\ge 0}$ with iid increments $X_{1},X_{2},\ldots$ and the existence of moments of various related quantities like the first passage into $[x,\infty)$ and the last exit time from $(-\infty,x]$ for arbitrary $x\in\mathbb{R}_{\geqslant}$ are studied in the Markov-modulated situation when the $X_{n}$ are governed by a positive recurrent Markov chain $M=(M_{n})_{n\ge 0}$ on a countable state space $\mathcal{S}$, thus for a Markov random walk $(M_{n},S_{n})_{n\ge 0}$. Our approach is based on the natural strategy to draw on the results in the iid case for the embedded ordinary random walks $(S_{\tau_{n}(i)})_{n\ge 0}$, where $\tau_{1}(i),\tau_{2}(i),\ldots$ denote the successive return times of $M$ to state $i$, and an analysis of the excursions of the walk between these epochs. However, due to these excursions, generalizations of the afore-mentioned theorems are not one-to-one extensions of those in the iid case and cannot be as illustrated by a number of counterexamples. In fact, various excursion measures will have to be introduced so as to characterize the existence of moments of different quantities.