On invariant measures of stochastic recursions in a critical case

Dariusz Buraczewski

doi:10.1214/105051607000000140

On invariant measures of stochastic recursions in a critical case

Probability 2007-10-25 v1

Authors: Dariusz Buraczewski

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Abstract

We consider an autoregressive model on $\mathbb{R}$ defined by the recurrence equation $X_n=A_nX_{n-1}+B_n$ , where $\{(B_n,A_n)\}$ are i.i.d. random variables valued in $\mathbb{R}\times\mathbb{R}^+$ 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 $\{X_n\}$ . 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.

Keywords

invariant measures stochastic recursions autoregressive model critical case radon measure stationary measures queuing theory

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@article{arxiv.0710.3687,
  title  = {On invariant measures of stochastic recursions in a critical case},
  author = {Dariusz Buraczewski},
  journal= {arXiv preprint arXiv:0710.3687},
  year   = {2007}
}

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Published in at http://dx.doi.org/10.1214/105051607000000140 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

On invariant measures of stochastic recursions in a critical case

Abstract

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