Stability of perpetuities in Markovian environment

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.