On invariant measures of stochastic recursions in a critical case

  title={On invariant measures of stochastic recursions in a critical case},
  author={Dariusz Buraczewski},
  journal={Annals of Applied Probability},
  • D. Buraczewski
  • Published 1 August 2007
  • Mathematics
  • Annals of Applied Probability
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… 
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  • (Russian) Litovsk. Mat. Sb. 15
  • 1975
Principles of Random Walk. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London
  • MR0171290 Institute of Mathematics University of Wroc law pl. Grunwaldzki
  • 1964
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