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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.

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