## The Annals of Probability

### Asymptotics for hitting times

#### Abstract

In this paper we characterize possible asymptotics for hitting times in aperiodic ergodic dynamical systems: asymptotics are proved to be the distribution functions of subprobability measures on the line belonging to the functional class $$\text{(A)}\qquad \mathcal{F}=\left\{F: \mathbb R\to [0,1]: \left\lbrack \matrix{ F\text{ is increasing, null on ]−∞,0];}\hfill \cr F\text{ is continuous and concave;}\hfill \cr F(t)\le t\text{ for }t\ge 0.\hfill}\right.\right\}.$$ Note that all possible asymptotics are absolutely continuous.

#### Article information

Source
Ann. Probab., Volume 33, Number 2 (2005), 610-619.

Dates
First available in Project Euclid: 3 March 2005

https://projecteuclid.org/euclid.aop/1109868594

Digital Object Identifier
doi:10.1214/009117904000000883

Mathematical Reviews number (MathSciNet)
MR2123204

Zentralblatt MATH identifier
1065.37006

#### Citation

Kupsa, M.; Lacroix, Y. Asymptotics for hitting times. Ann. Probab. 33 (2005), no. 2, 610--619. doi:10.1214/009117904000000883. https://projecteuclid.org/euclid.aop/1109868594

#### References

• Abadi, M. (2004). Sharp error terms and necessary conditions for exponential hitting times in mixing processes. Ann. Probab. 32 243–264.
• Abadi, M. and Galves, A. (2001). Inequalities for the occurrence times of rare events in mixing processes. The state of art. Markov Process. Related Fields 7 97–112.
• Chazottes, J.-R. (2003). Hitting and returning to non-rare events in mixing dynamical systems. Nonlinearity 16 1017–1034.
• Coelho, Z. (2000). Asymptotic laws for symbolic dynamical systems. Topics in Symbolic Dynamics and Applications. Lecture Notes in Math. 279 123–165. Springer, New York.
• Coelho, Z. and de Faria, E. (1990). Limit laws for entrance times for homeomorphisms of the circle. Israel J. Math. 69 235–249.
• Collet, P. and Galves, A. (1993). Statistics of close visits to the indifferent fixed point of an interval map. J. Statist. Phys. 72 459–478.
• Geman, D. (1973). A note on the distribution of hitting times. Ann. Probab. 1 854–856.
• Hirata, M., Saussol, B. and Vaienti, S. (1999). Statistics of return times: A general framework and new applications. Comm. Math. Phys. 206 33–55.
• Kac, M. (1947). On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc. 53 1002–1010.
• Lacroix, Y. (2002). Possible limit laws for entrance times of an ergodic aperiodic dynamical system. Israel J. Math. 132 253–264.
• Saussol, B. (1998). Etude statistique de systèmes dynamiques dilatants. Ph.D. thesis, Toulon, France.
• Shields, P. (1973). The Theory of Bernoulli Shifts. Univ. Chicago Press, Illinois.
• Young, L. S. (1999). Recurrence times and rates of mixing. Israel J. Math. 110 153–188.