On thedimension of the predictive process of a memoryless channel

Abstract

We analyze metrical properties of the unique stationary law for the one-step predictor of a finite state Markov Chain from noisy observations. In Piccioni (1990), the topological aspect of this problem was analyzed. Our work is a natural follow-up of this paper. We will be concerned with the case where the stationary law has support in a totally disconnected and perfect set. In this case the predictor keeps an infinite memory of the past observations. The closure of the support of this stationary law is called the attractor S(Elton and Piccioni (1992)). We present a lower bound for the dimension of S. This lower bound will be also an upper bound for the exponent scale of the law. As a consequence of our results, we partially answer a question raised by Piccioni (1990), in a case (b=c, see notation in Section 4) where the closure of the invariant measure's support is an interval, showing that the stationary law is singular with respect to the Lebesgue measure