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- W2544216139 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 $(A_{n},B_{n})_{nge 1}$ is modulated by an ergodic Markov chain $(M_{n})_{nge 0}$ 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 $Psi_{1}circldotscircPsi_{n}(Z_{0})$ 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 $Psi_{n}circldotscircPsi_{1}$. 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 $c_{i}$, $iinmathcal{S}$, 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 $cinmathbb{R}$, and (3) forward and backward iterations generally have different laws given $M_{0}=i$ for $iinmathcal{S}$ so that the former ones need a separate analysis. Our proofs draw on related results for the iid-case, notably by Vervaat, Grinceviv{c}ius, and Goldie and Maller, in combination with recent results by the authors on fluctuation theory for Markov random walks." @default.
- W2544216139 created "2016-11-04" @default.
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- W2544216139 date "2017-01-20" @default.
- W2544216139 modified "2023-09-26" @default.
- W2544216139 title "Stability of perpetuities in Markovian environment" @default.
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- W2544216139 doi "https://doi.org/10.1080/10236198.2016.1271878" @default.
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