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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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[1] Arjas, E.; Speed, T.P., An extension of cramér’s estimate for the absorption probability of a random walk, Proc. camb. phil. soc., 73, 355-359, (1973) · Zbl 0254.60041
[2] Asmussen, S., Aspects of matrix wiener – hopf factorisation in applied probability, Math. scientist, 14, 101-116, (1989) · Zbl 0688.60055
[3] Asmussen, S., Applied probability and queues, (2003), Springer New York · Zbl 1029.60001
[4] Athreya, K.B.; Rama Murthy, K., Feller’s renewal theorem for systems of renewal equations, J. Indian inst. sci., 58, 437-459, (1976) · Zbl 0356.60057
[5] Brandt, A., The stochastic equation \(Y_{n + 1} = A_n Y_n + B_n\) with stationary coefficients, Adv. appl. probab., 18, 211-220, (1986)
[6] Chow, S.C.; Teicher, H., Probability theory. independence, interchangeability, martingales, (1978), Springer New York, Heildelberg, Berlin · Zbl 0399.60001
[7] Crump, K.S., On systems of renewal equations, J. math. anal. appl., 30, 425-434, (1970) · Zbl 0198.22502
[8] Feller, W., Introduction to probability theory and its applications, (1971), Wiley New York, London, Sydney · Zbl 0219.60003
[9] Goldie, C.M., Implicit renewal theory and tails of solutions of random equations, Ann. appl. probab., 1, 26-166, (1991) · Zbl 0724.60076
[10] Grincevičius, A.K., Products of random affine transformations, Lithuanian math. J., 20, 279-282, (1980) · Zbl 0472.60035
[11] Hamilton, J.D., Estimation, inference and forecasting of time series subject to change in regime, (), 230-260
[12] Horn, R.; Johnson, C., Matrix analysis, (1985), Cambridge University Press Cambridge · Zbl 0576.15001
[13] Kesten, H., Random difference equations and renewal theory for products of random matrices, Acta math., 131, 207-248, (1973) · Zbl 0291.60029
[14] Kesten, H., Renewal theory for functionals of a Markov chain with general state space, Ann. probab., 2, 355-386, (1974) · Zbl 0303.60090
[15] C. Klüppelberg, S. Pergamenchtchikov, The tail of the stationary distribution of a random coefficient AR(q) model, Ann. Appl. Probab. 14 (2004) 971-1005. · Zbl 1094.62114
[16] E. Le Page, Théorèmes de renouvellement pour les produits de matrices aléatoires. Equations aux différences aléatoires, Séminaires de probabilités de Rennes, 1983.
[17] de Saporta, B., Renewal theorem for a system of renewal equations, Ann. inst. H. poincare probab. statist., 39, 823-838, (2003) · Zbl 1021.60069
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