NDLI: Poisson and compound Poisson approximations in conventional and nonconventional setups

Content Provider	Springer Nature Link
Author	Rapaport, Ariel Kifer, Yuri
Copyright Year	2013
Abstract	It was shown in Kifer (Israel J Math, 2013) that for any subshift of finite type considered with a Gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically Poisson distributed. Here we not only extend this result to all $$\psi $$ -mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to Poisson or to compound Poisson distributions. It turns out that for all nonperiodic points a limiting distribution is always Poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is Bernoulli in which case the limiting distribution always exists. Thus we describe, essentially completely, limiting distributions of multiple recurrence numbers in this setup. As a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed. Most of the results are new also for the widely studied single recurrencies case (see, for instance, Haydn and Vaienti Discret Contin Dyn Syst A 10:589–616, 2004; Probab Theory Relat Fields 144:517–542, 2009; Abadi and Saussol Stoch Process Appl 121:314–323, 2011; Abadi and Vergne Nonlinearity 21:2871–2885, 2008), as well.
Ending Page	831
Page Count	35
Starting Page	797
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	3
Volume Number	160
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-11-10
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	$$\psi $$ ψ -mixing Mathematical and Computational Biology Theoretical, Mathematical and Computational Physics Probability Theory and Stochastic Processes Quantitative Finance Thermodynamic formalism, variational principles, equilibrium states Subshift of finite type Statistics for Business/Economics/Mathematical Finance/Insurance Operations Research/Decision Theory Nonconventional sums Central limit and other weak theorems Discrete-time Markov processes on general state spaces Poisson and compound Poisson appproximations
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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