Miguel Abadi

A version of Maurer's conjecture for stationary ψ-mixing processes

For a stationary source with finite alphabet, let Rn be the number of nonoverlapping n-blocks of symbols, occurring before the initial n-block reappears. When the source is ψ-mixing, we prove that the difference between the expectation of log Rn 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 J. Cryptol. 5 89–105). Our theorem generalizes recent results presented in Coron and Naccache (1999LectureNotes in Computer Science vol 1556, pp 51–71), Choe and Kim (2000 Coll. Math. 84 159–71) and Wegenkittl (2001 IEEE Trans. Inform. Theory 47 2480–9), in the context of Markov chains. We also prove that the difference between the variance of log Rn and the variance of the probability of 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 distribution of the number of non-overlapping n-blocks until a fixed but otherwise arbitrary nblock reappears. This is a new result that is interesting by itself.

The distribution of the short-return function

We consider the first return of a sequence to itself . It is known that Tn/n converges to 1 almost surely for ergodic processes with positive entropy. Large deviations are also known. We study the fluctuations of Tn. For product measures over a countable alphabet we prove the convergence in distribution of Tn, properly re-scaled, to a non-degenerated, non-parametric, distribution. We also show that the re-scaled factors behave as follows: expectation of Tn is about n minus a positive constant and the variance is about a constant. As an application, we compute bounds and limiting values of the proportion of self-avoiding sequences. We illustrate with some examples.

Almost sure convergence of the clustering factor in α-mixing processes

Abadi and Saussol (2011) have proved that the first time a dynamical system, starting from its equilibrium measure, hits a target set A has approximately an exponential law. These results hold for systems satisfying the α-mixing condition with rate function α decreasing to zero at any rate. The parameter of the exponential law is the product λ(A)μ(A), where the latter is the measure of the set A; only bounds for λ(A) were given. In this note we prove that, if the rate function α decreases algebraically and if the target set is a sequence of nested cylinders sets An(x) around a point x, then λ(An) converges to one for almost every point x. As a byproduct, we obtain the corresponding result for return times.

Rényi Entropies and Large Deviations for the First Match Function

We define the first match function Tn : Cⁿ → {1, ... , n} where C is a finite alphabet. For two copies of x₁ⁿ ∈ Cⁿ, this function gives the minimum number of steps one has to slide one copy of x₁ⁿ to get a match with the other one. For ergodic positive entropy processes, Saussol and coauthors proved the almost sure convergence of T_n/n. We compute the large deviation properties of this function. We prove that this limit is related to the Rényi entropy function, which is also proved to exist. Our results hold under a condition easy to check which defines a large class of processes. We provide some examples.