Probabilistic shadowing in linear skew products

We investigate the probability of the event that a finite random pseudotrajectory can be effectively shadowed by an exact trajectory. The main result of the work describes a class of skew products, for which this probability tends to one as the length of a pseudotrajectory tends to infinity and the value of a maximal mistake on each step tends to zero. We also show that continuous linear skew products over a Bernoulli shift, doubling map on a circle and any Anosov linear map on a torus lie in this class. The Cramer’s large deviation theorem is used in the proof.