A Cassels-Schmidt theorem for non-homogeneous Markov chains

Let r be an integer not less than 2. Suppose that we have a (not necessarily homogeneous) Markov chain with state space {0,1,…,r−1} given by the sequence of r×r transition matrices P(n)=(pij(n)),i,j∈{0,1,…,r−1},n=1,2,…. The chain above generates a probability measure μ on [0,1]. We prove that if inf{pij(n),i,j=0,1,…,r−1,n∈N}>0 and if s>1 is an algebraic number such that the ratio logs/logr is irrational, then μ-almost all the numbers are normal to base s. This generalizes a result of Puskhin [Theory Probab. Appl. 41 (3) (1996) 593-597]. We also estimate the Hausdorff dimension of sets of numbers which are determined in terms of the frequencies of their r-adic digits and are normal to base s. This result extends those of Volkmann [Bull Sci. Math. II Ser. 108 (1984) 321-336; 109 (1985) 209-223] and Billingsley [Ergodic Theory and Information, Wiley, New York, 1965].