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January 1997 The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case
Martine Babillot, Philippe Bougerol, Laure Elie
Ann. Probab. 25(1): 478-493 (January 1997). DOI: 10.1214/aop/1024404297


Let $(B_n, A_n)_{n \geq 1}$ be a sequence of i.i.d. random variables with values in $\mathbf{R}^d \times \mathbf{R}_*^+$. The Markov chain on $\mathbf{R}^d$ which satisfies the random equa tion $X_n = A_n X_{n-1} + B_n$ is studied when $\mathbf{E}(\log A_1) = 0$. No density assumption on the distribution of $(B_1, A_1)$ is made. The main results are recurrence of the Markov chain $X_n$, stability properties of the paths, existence and uniqueness of a Radon invariant measure and a limit theorem for the occupation times. The results rely on a renewal theorem for the process $(X_n, A_n \dots A_1)$.


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Martine Babillot. Philippe Bougerol. Laure Elie. "The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case." Ann. Probab. 25 (1) 478 - 493, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0873.60045
MathSciNet: MR1428518
Digital Object Identifier: 10.1214/aop/1024404297

Primary: 60J10
Secondary: 60B15, 60J15, 60K05

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 1 • January 1997
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