Let @math be a product of @math independent, identically distributed random matrices @math , with the properties that @math is bounded in @math , and that @math has a deterministic (constant) invariant vector. Assuming that the probability of @math having only the simple eigenvalue 1 on the unit circle does not vanish, we show that @math is the sum of a fluctuating and a decaying process. The latter converges to zero almost surely, exponentially fast as @math . The fluctuating part converges in Cesaro mean to a limit that is characterized explicitly by the deterministic invariant vector and the spectral data of @math associated to 1. No additional assumptions are made on the matrices @math ; they may have complex entries and not be invertible. We apply our general results to two classes of dynamical systems: inhomogeneous Markov chains with random transition matrices (stochastic matrices), and random repeated interaction quantum systems. In both cases, we prove ergodic theorems for the dynamics, and we obtain the form of the limit states.
Getting informations on the fluctuations of the process around its limiting value is certainly an interesting and important issue. It amounts to getting informations about the law of the vector valued random variable @math of Theorem , which is quite difficult in general. There are recent partial results about aspects of the law of such random vectors in case they are obtained by means of matrices belonging to some subgroups of @math satisfying certain irreducibility conditions, see e.g. @cite_20 . However, these results do not apply to our situation.
@cite_20
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Abstract We study the behavior at infinity of the tail of the stationary solution of a multidimensional linear auto-regressive process with random coefficients. We exhibit an extended class of multiplicative coefficients satisfying a condition of irreducibility and proximality that yield to a heavy tail behavior. To cite this article: B. de , C. R. Acad. Sci. Paris, Ser. I 339 (2004).