Hitting and return times in ergodic dynamical systems
Haydn, N. ; Lacroix, Y. ; Vaienti, S.
Ann. Probab., Tome 33 (2005) no. 1, p. 2043-2050 / Harvested from Project Euclid
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Given an ergodic dynamical system (X,T,μ), and U⊂X measurable with μ(U)>0, let μ(U)τU(x) denote the normalized hitting time of x∈X to U. We prove that given a sequence (Un) with μ(Un)→0, the distribution function of the normalized hitting times to Un converges weakly to some subprobability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution function F̃, and that in the converging case, ¶ \[(\diamondsuit)\hspace*{66pt}F(t)=\int_{0}^{t}\bigl(1-\tilde{F}(s)\bigr)\,ds,\qquad t\ge0.\hspace*{66pt}\] ¶ This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also.
Publié le : 2005-09-14
Classification:  Asymptotic distribution,  hitting,  return times,  Kac,  37A05,  37A50,  60F05,  28D05 
@article{1127395881,
     author = {Haydn, N. and Lacroix, Y. and Vaienti, S.},
     title = {Hitting and return times in ergodic dynamical systems},
     journal = {Ann. Probab.},
     volume = {33},
     number = {1},
     year = {2005},
     pages = { 2043-2050},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1127395881}
}

Haydn, N.; Lacroix, Y.; Vaienti, S. Hitting and return times in ergodic dynamical systems. Ann. Probab., Tome 33 (2005) no. 1, pp.  2043-2050. http://gdmltest.u-ga.fr/item/1127395881/