Geometric law for multiple returns until a hazard

摘要

For a psi-mixing stationary process xi(0), xi(1), xi(2), ... we consider the number N-N of multiple recurrencies {xi(qi()(n)) is an element of Gamma(N), i = 1, ..., l} to a set Gamma(N) for n until the moment tau(N) (which we call a hazard) when another multiple recurrence {xi(qi)((n)) is an element of Delta(N), i = 1, ..., l} takes place for the first time where Gamma(N) boolean AND Delta(N) = empty set and q(i)(n) < q(i+1)(n), i = 1, ..., l are nonnegative increasing functions taking on integer values on integers. It turns out that if P{xi(0) is an element of Gamma(N)} and P{xi(0) is an element of Delta(N)} decay in N with the same speed then N-N converges weakly to a geometrically distributed random variable. We obtain also a similar result in the dynamical systems setup considering a psi-mixing shift T on a sequence space Omega and study the number of multiple recurrencies {T-qi(n)omega is an element of A(n)(b), i = 1, ..., l} until the first occurence of another multiple recurrence {T-qi(n)omega is an element of A(m)(a), i = 1, ...,l} where A(m)(a), A(n)(b) are cylinder sets of length m and n constructed by sequences a, b is an element of Omega 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'.