The Annals of Applied Probability

On invariant measures of stochastic recursions in a critical case

Dariusz Buraczewski

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 10 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1186755239

Digital Object Identifier
doi:10.1214/105051607000000140

Mathematical Reviews number (MathSciNet)
MR2344306

Zentralblatt MATH identifier
1151.60034

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1186755239


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References

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