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

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. When the system is driven by an i.i.d sequence, we prove that the memory loss property is generic in the following sense : if it is not fulfilled, the support of the common law of the sequence is included in a finite union of affine hyperplanes and in the discrete case the atoms of the measure are linearly related.