Hitting and return times in ergodic dynamical systems

By N. Haydn, Y. Lacroix, Sandro Vaienti. 2004

110 citations recorded by the source corpus.

Given an ergodic dynamical system (X, T,μ), and U C X measurable with μ(U) > 0, let μ(U)τ U (x) denote the normalized hitting time of x ∈ X to U. We prove that given a sequence (U n ) with μ(U n ) → 0, the distribution function of the normalized hitting times to U n 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 F, and that in the converging case, (=) F(t)= ∫ 0 t (1-F(s))ds, t ≥ 0. 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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dbb606b773df79b931d6860a3a4e4c7c778ce264

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Mike Todd
2. A. C. Freitas
3. J. Freitas
4. Sandro Vaienti
5. N. Haydn

In Mathematical Physics

1. N. Haydn
2. J. Freitas
3. Mike Todd
4. A. C. Freitas
5. Sandro Vaienti
6. Matthew Nicol
7. Andrei Török

Citations in the corpus

• The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics (2012, methodology): “TION IN NON-UNIFORMLY HYPERBOLIC SYSTEMS 5 The existence of exponential HTS (G(t) = 1−e−t) is equivalent to the existence of exponential RTS (G˜(t) = 1−e−t). In fact, according to the Main Theorem in [HLV05], a system has HTS Gif and only if it has RTS G˜ and G(t) = Zt 0 (1 −G˜(s))ds. (1.4) In [FFT1”
• Leaking chaotic systems (2012, background): “The times Ti’s between two consecutive returns are called Poincaré recurrence times, 14 a central concept in dynamical-systems theory (Haydn et al., 2005).”
• AN INTRODUCTION TO QUANTITATIVE POINCARÉ RECURRENCE IN DYNAMICAL SYSTEMS (2009, unknown): “On the other side, if a limiting distribution exists for the return time, then it also exists for the hitting time, and the two are related by an integral relation [31].”
• Entry and return times distribution (2013, unknown): “Theorem 4 ([5,6]): Let Bn ⊂ (μ(Bn) > 0) be a sequence of sets so that μ(Bn) → 0+ .”
• The compound Poisson distribution and return times in dynamical systems (2008, background): “In the case of the first entry and return times, these two distributions have for general ergodic systems been linked in [12].”