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

Abstract

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.

Potential Well in Poincaré Recurrence

“…Abadi and Saussol [ 29 ], in the more general case known up to now, proved that for -mixing processes with at least polynomially decaying rates, the distribution of the hitting and return time converges, almost surely, to an exponential with parameter of one. We refer to [ 8 ] for the precise definition of -mixing, but the only important point for us it to know that summable -mixing implies -mixing with at least polynomially decaying rates.…”

“…Finally, by doing t = r in (29) and applying the MVT once again, we get: µ(T A > r) − e −ρ(A)µ(A)(r−τ(A))…”

Potential well in Poincaré recurrence

“…Abadi and Saussol [AS16], in the more general case known up to now, proved that for α-mixing processes with at least polynomially decaying α rates, the distribution of hitting and return time converge, almost surely, to an exponential with parameter 1. We refer [Bra05] for the precise definition of α-mixing, but the only important point for us it to know that summable φ-mixing implies α-mixing with at least polynomially decaying α rates.…”

Matching of observations of dynamical systems, with applications to sequence matching

“…We write D q = D Id q , the generalized dimension of order q of the original measure. 1 it could be a discretized version of a ow 2…”