The compound Poisson distribution and return times in dynamical systems

• Published in: Probability theory and related fields Vol. 144; no. 3-4; pp. 517 - 542
• Main Authors: Haydn, Nicolai, Vaienti, Sandro
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.07.2009; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Constraining; Distribution theory; Dynamical Systems; Economics; Exact sciences and technology; Finance; General topics; Insurance; Limit theorems; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Poisson distribution; Probability; Probability and statistics; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Sciences and techniques of general use; Statistics for Business; Studies; Theoretical; Theory; Secondary: 62E17; Primary: 37A50; Mixed distribution; Time distribution; Mixing; Error estimation; Limit distribution; Probability theory; Distribution function; Dynamical system; Poisson distribution; Convergence; recurrence; mixing
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-008-0153-y

Previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere Poissonian. Here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound Poissonian distribution. We also derive error terms for the convergence to the limiting distribution. We also prove a very general theorem that can be used to establish compound Poisson distributions in many other settings.