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The effect of reducible Markov modulation on tail probabilities in models of random growth

Brendan K. Beare, Alexis Akira Toda

TL;DR

The paper extends tail-probability results for Markov-modulated random growth from irreducible to reducible Markov modulation. By formulating the tail problem in terms of a holomorphic matrix-valued function $A(z)$ and analyzing the pole structure of $A(z)^{-1}$ through class- and chain-based decomposition, it shows that the upper tail of the log-wealth distribution is Erlang-like with shape equal to the longest chain of classes $d$ and rate $$ determined by the right abscissa where $$ solves $= ext{the zero condition }= ext{right abscissa of }A(s)$. The results leverage Laplace transforms and a Rothblum-type index theorem to connect tail behavior to the combinatorial structure of the Markov modulator, and apply the theory to stopped Markov additive processes and discrete-time processes with reset, yielding practical Erlang-branch tail characterizations for a broad class of economic growth models.

Abstract

A recent economic literature deals with models of random growth in which the size of economic agents is subject to light-tailed Markov-modulated additive shocks. It has been shown that if Markov modulation is irreducible then the models give rise to a distribution of sizes across agents whose upper tail resembles, in a particular sense, that of an exponential distribution. We show that while this need not be the case under reducible Markov modulation, the upper tail will nevertheless resemble that of an Erlang distribution. A novel extension of the Rothblum index theorem allows us to characterize the Erlang shape parameter.

The effect of reducible Markov modulation on tail probabilities in models of random growth

TL;DR

The paper extends tail-probability results for Markov-modulated random growth from irreducible to reducible Markov modulation. By formulating the tail problem in terms of a holomorphic matrix-valued function and analyzing the pole structure of through class- and chain-based decomposition, it shows that the upper tail of the log-wealth distribution is Erlang-like with shape equal to the longest chain of classes and rate determined by the right abscissa where solves . The results leverage Laplace transforms and a Rothblum-type index theorem to connect tail behavior to the combinatorial structure of the Markov modulator, and apply the theory to stopped Markov additive processes and discrete-time processes with reset, yielding practical Erlang-branch tail characterizations for a broad class of economic growth models.

Abstract

A recent economic literature deals with models of random growth in which the size of economic agents is subject to light-tailed Markov-modulated additive shocks. It has been shown that if Markov modulation is irreducible then the models give rise to a distribution of sizes across agents whose upper tail resembles, in a particular sense, that of an exponential distribution. We show that while this need not be the case under reducible Markov modulation, the upper tail will nevertheless resemble that of an Erlang distribution. A novel extension of the Rothblum index theorem allows us to characterize the Erlang shape parameter.
Paper Structure (9 sections, 18 theorems, 70 equations, 1 figure)

This paper contains 9 sections, 18 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The irreducible case
  3. The general case
  4. Classes and chains of classes
  5. Main result
  6. The Rothblum index theorem
  7. Applications to random growth models
  8. Stopped Markov additive process
  9. Discrete-time Markov additive process with reset

Key Result

Theorem 1.1

Let $X$ be a real random variable and $\varphi$ its Laplace transform, with right abscissa of convergence $\alpha\in(0,\infty)$. Suppose that $\varphi^\circ$ can be meromorphically extended to an open set containing $\alpha$ in such a way that $\alpha$ is a pole of this extension. Let $d$ be the ord

Figures (1)

  • Figure 1: The directed graph $G(A)$ for the matrix $A$ in \ref{['eq:exampleA']}.

Theorems & Definitions (26)

  • Theorem 1.1: Graham-Vaaler-Nakagawa
  • Proposition 2.1
  • Lemma 2.1: Perron-Frobenius
  • Lemma 2.2
  • Lemma 2.3: Keldysh
  • proof : Proof of Proposition \ref{['thm:irreducible']}
  • Theorem 3.1
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['thm:main']}
  • ...and 16 more