Stability of perpetuities in Markovian environment

TL;DR

This paper investigates the stability and limit laws of affine linear iterations in a Markovian environment, providing necessary and sufficient conditions for convergence and describing the dependence of limit laws on the initial state of the Markov chain.

Contribution

It extends the analysis of perpetuities to Markov-modulated environments, characterizing convergence and limit laws with new conditions and highlighting differences from the iid case.

Findings

Backward iterations may converge in distribution even if almost sure convergence fails.

The degenerate case in Markovian settings is more complex than in iid cases.

Forward and backward iterations have different laws conditioned on the initial Markov state.

Abstract

The stability of iterations of affine linear maps $\Psi_{n}(x)=A_{n}x+B_{n}$, $n=1,2,\ldots$, is studied in the presence of a Markovian environment, more precisely, for the situation when $(A_{n},B_{n})_{n\ge 1}$ is modulated by an ergodic Markov chain $(M_{n})_{n\ge 0}$ with countable state space $\mathcal{S}$ and stationary distribution $\pi$. We provide necessary and sufficient conditions for the a.s. and the distributional convergence of the backward iterations $\Psi_{1}\circ\ldots\circ\Psi_{n}(Z_{0})$ and also describe all possible limit laws as solutions to a certain Markovian stochastic fixed-point equation. As a consequence of the random environment, these limit laws are stochastic kernels from $\mathcal{S}$ to $\mathbb{R}$ rather than distributions on $\mathbb{R}$, thus reflecting their dependence on where the driving chain is started. We give also necessary and sufficient conditions for the distributional convergence of the forward iterations $\Psi_{n}\circ\ldots\circ\Psi_{1}$. The main differences caused by the Markovian environment as opposed to the extensively studied case of independent and identically distributed (iid) $\Psi_{1},\Psi_{2},\ldots$ are that: (1) backward iterations may still converge in distribution, if a.s. convergence fails, (2) the degenerate case when $A_{1}c_{M_{1}}+B_{1}=c_{M_{0}}$ a.s. for suitable constants $c_{i}$, $i\in\mathcal{S}$, is by far more complex than the degenerate case for iid $(A_{n},B_{n})$ when $A_{1}c+B_{1}=c$ a.s. for some $c\in\mathbb{R}$, and (3) forward and backward iterations generally have different laws given $M_{0}=i$ for $i\in\mathcal{S}$ so that the former ones need a separate analysis. Our proofs draw on related results for the iid-case, notably by Vervaat, Grincevi\v{c}ius, and Goldie and Maller, in combination with recent results by the authors on fluctuation theory for Markov random walks.