The Annals of Applied Probability

On invariant measures of stochastic recursions in a critical case

Dariusz Buraczewski

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We consider an autoregressive model on ℝ defined by the recurrence equation Xn=AnXn−1+Bn, where {(Bn, An)} are i.i.d. random variables valued in ℝ×ℝ+ and $\mathbb{E}[\log A_{1}]=0$ (critical case). It was proved by Babillot, Bougerol and Elie that there exists a unique invariant Radon measure of the process {Xn}. The aim of the paper is to investigate its behavior at infinity. We describe also stationary measures of two other stochastic recursions, including one arising in queuing theory.

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Ann. Appl. Probab., Volume 17, Number 4 (2007), 1245-1272.

First available in Project Euclid: 10 August 2007

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Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60G50: Sums of independent random variables; random walks

Random coefficients autoregressive model affine group random equations queues contractive system regular variation


Buraczewski, Dariusz. On invariant measures of stochastic recursions in a critical case. Ann. Appl. Probab. 17 (2007), no. 4, 1245--1272. doi:10.1214/105051607000000140.

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