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 $(X_n, A_n \dots A_1)$.
Publié le : 1997-01-14
Classification:
Random coefficient autoregressive models,
random walk,
affine group,
stability,
renewal theorem,
limit theorem,
60J10,
60K05,
60B15,
60J15
@article{1024404297,
author = {Babillot, Martine and Bougerol, Philippe and Elie, Laure},
title = {The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in
the critical case},
journal = {Ann. Probab.},
volume = {25},
number = {4},
year = {1997},
pages = { 478-493},
language = {en},
url = {http://dml.mathdoc.fr/item/1024404297}
}
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., Tome 25 (1997) no. 4, pp. 478-493. http://gdmltest.u-ga.fr/item/1024404297/