Poisson and compound Poisson approximations in conventional and nonconventional setups

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.