Potential Well Spectrum and Hitting Time in Renewal Processes

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

“…This is of particular interest for using with a small value of m. In particular, one may take m equal to the period of a periodic point x when A is the n-cylinder around it, say T (A). Thus, we generalize the results on [1,2,5] where under more restrictive conditions, it was proved that |λ(…”

“…We emphasize that this formula is sharp for the binary renewal stochastic process [2]. In particular this holds for the Markov chain model of the Pommeau-Manneville intermittent map.…”

“…The proof is identical to that of Theorem 3.1 in [2] changing both T (A x n ) and n by N n , and s by S n , as we did here in Theorem 2.2. Actually we only need to show the approximation between λ(A x n ) and λ 3 (A x n ) defined there.…”

“…Following [2] we show that it can be well approximated by the arithmetic mean of escape time probabilities. These escape times range between two appropriate explicit scales.…”

“…We recall the reader's attention to a recent paper by Abadi, Cardeño and Gallo [2] where it is described the spectrum of values of this parameter in the whole phase space for positive recurrent renewal processes. The authors prove that, in that case, the parameter λ(A) represents the probability the process has to escape the set A very soon after having entered A.…”