The Annals of Probability

The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case

Martine Babillot, Philippe Bougerol, and Laure Elie

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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 $(X_n, A_n \dots A_1)$.

Article information

Ann. Probab., Volume 25, Number 1 (1997), 478-493.

First available in Project Euclid: 18 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K05: Renewal theory 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60J15

Random coefficient autoregressive models random walk affine group stability renewal theorem limit theorem


Babillot, Martine; Bougerol, Philippe; Elie, Laure. The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case. Ann. Probab. 25 (1997), no. 1, 478--493. doi:10.1214/aop/1024404297.

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