Geometric Law for Multiple Returns until a Hazard
Abstract
For a -mixing stationary process 0,1,2,... we consider the number NN of multiple recurrencies \qi(n)∈N,\, i=1,...,\ to a set N for n until the moment τN (which we call a hazard) when another multiple recurrence \qi(n)∈N,\, i=1,...,\ takes place for the first time where NN= and qi(n)<qi+1(n),\, i=1,..., are nonnegative increasing functions taking on integer values on integers. It turns out that if P\0∈N\ and P\0∈N\ decay in N with the same speed then NN converges weakly to a geometrically distributed random variable. We obtain also a similar result in the dynamical systems setup considering a -mixing shift T on a sequence space and study the number of multiple recurrencies \ Tqi(n)ω∈ Anb,\, i=1,...,\ until the first occurence of another multiple recurrence \ Tqi(n)ω∈ Ama,\, i=1,...,\ where Ama,\, Anb are cylinder sets of length m and n constructed by sequences a,b∈, respectively, and chosen so that their probabilities have the same order. This work is motivated by a number of papers on asymptotics of numbers of single and multiple returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a "hole".