Memory loss property for products of random matrices in the $(\max,+)$ algebra

Abstract

Products of random matrices in the $(\max,+)$ algebra are used as a model for a class of discrete event dynamical systems. J. Mairesse proved that such a system couples in finite times with a unique stationary regime if and only if it has a memory loss property. We prove that the memory loss property is generic in the following sense : if it is not fulfilled, the support of the measure is included in a finite union of affine hyperplanes and in the discrete case the atoms of the measure are linearly related.