A correct version of Maurer's conjecture for $\psi$-mixing processes (72K, dvi) Mar 14, 03
Abstract. We consider the number of non-overlapping blocks of $n$ symbols occuring before the initial $n$-block reappears. For $\psi$-mixing processes on a finite alphabet, we prove that the difference between the expectation of the logarithm of this number and the entropy of $n$-blocks converges to the constant of Euler divided by $-\ln(2)$. This can be considered the correct version of a conjecture presented in Maurer (1992). Our theorem generalizes recent results presented in Coron and Nacache (1999), Choe and Kim (2000) and Wegenkittl (2001), in the context of Markov chains. We also prove that the difference between the variance of the logarithm of this number and the variance of the law of the $n$-blocks converges to an explicit constant as $n$ diverges. The basic ingredient of the proofs is an upper-bound for the exponential approximation of the law of the number of non-overlapping $n$-blocks until a fixed but otherwise arbitrary $n$-block reappears. This is a new result which is interesting by itself.