Geometric law for multiple returns until a hazard

For a ψ-mixing stationary process {equation presented} we consider the number NN of multiple recurrencies {equation presented} to a set lN for n until the moment lN (which we call a hazard) when another multiple recurrence {equation presented} l are nonnegative increasing functions taking on integer values on integers. It turns out that if P {equation presented} 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 ψ-mixing shift T on a sequence space T and study the number of multiple recurrencies {equation presented} until the first occurence of another multiple recurrence {equation presented} where A ^a _m , A ^b _n are cylinder sets of length m and n constructed by sequences a, b Ω, 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'.