Normalized image of a vector by an infinite product of nonnegative matrices

A sofic measure is the image of a Markov probability measure by a continuous morphism, and can be represented by means of products of matrices $A_n$ that belong to a finite set of nonnegative matrices. To prove that the multifractal formalism holds for such a measure, it is necessary to know whenever the sequence $n\mapsto \frac{A_1\cdots A_{n}v}{\mathrm{\parallel }{A}_1\cdots A_{n}v\mathrm{\parallel }}$ converges when $v$ is a positive vector. We give a sufficient condition for this convergence, that we use for the study of one Bernoulli convolution.