M. Abadi, A. Galves
A correct version of Maurer's conjecture for $\psi$-mixing
processes
(72K, dvi)
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.