Reducible Markov modulation, pole order, and tail behavior in random growth models

TL;DR

This paper studies how reducible Markov modulation affects tail behavior in random growth models, showing it leads to Erlang-like tails and providing a mathematical characterization of this phenomenon.

Contribution

It introduces a theorem on the pole order of matrix functions with reducible Metzler structure, linking it to tail behavior in Markov-modulated growth models.

Findings

Reducible modulation results in Erlang-shaped tail distributions.

The main theorem characterizes the pole order of matrix functions with reducible Metzler structure.

Application of the theorem yields the Erlang shape parameter in specific growth models.

Abstract

Recent work on random growth models with light-tailed Markov-modulated additive shocks has shown that irreducible modulation yields tail behavior resembling an exponential distribution. We show that with reducible modulation the tail behavior more generally resembles an Erlang distribution. Our main technical contribution is a theorem on the order of a real pole of the inverse of a holomorphic matrix-valued function with reducible Metzler structure. In a special affine case, the theorem recovers the Rothblum index theorem. Applying this result together with a Tauberian theorem, we characterize the Erlang shape parameter in two models of Markov-modulated random growth.