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Tail of the stationary solution of the stochastic equation $$Y_{n+1}=a_{n} Y_{n}+b_{n}$$ with Markovian coefficients. (English) Zbl 1096.60025
The paper deals with the tail behaviour of the stationary solution $$Y_1$$ of the stochastic difference equation $$Y_{n+1}=a_n Y_n + b_n,\;n \in \mathbb Z$$, where the sequence $$(a_n)$$ is a real-valued finite state space Markov chain and $$(b_n)$$ is a sequence of i.i.d. real-valued r.v.’s independent of $$(a_n)$$. Further E $$\log| a_0| <0$$ and $$E \log^+| b_0| <\infty$$ are assumed ensuring the existence of a unique stationary solution $$(Y_n)$$. Under spectral radius properties related to the transition matrix on $$(a_n)$$, a technical assumption on the state space and a moment condition on $$b_0$$, it is proved that $$t^\lambda P(xY_1>t)\rightarrow L(x)$$ as $$t\rightarrow \infty$$, for $$x\in\{-1,1\}$$, where $$L(1)+L(-1)$$ is positive and $$\lambda > 0$$ is a parameter fulfilling an assumption. Similar results have already been proved in the case of i.i.d. matrices $$a_n$$ and $$Y_n,\,b_n$$ vectors by Kesten and Le Page. The proof is based on a new renewal theorem for systems of renewal equations as well as an extension of Grincevicius inequality (extension of Lévy’s symmetrization inequality) to the Markovian case, given in the paper.

##### MSC:
 60G99 Stochastic processes 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60K25 Queueing theory (aspects of probability theory)
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