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.
Categories: math.PR