Geometric Law for Multiple Returns until a Hazard

For a $\psi$-mixing stationary process $\xi_0,\xi_1,\xi_2,...$ we consider the number $\mathcal N_N$ of multiple recurrencies $\{\xi_{q_i(n)}\in\Gamma_N,\, i=1,...,\ell\}$ to a set $\Gamma_N$ for $n$ until the moment $\tau_N$ (which we call a hazard) when another multiple recurrence $\{\xi_{q_i(n)}\in\Delta_N,\, i=1,...,\ell\}$ takes place for the first time where $\Gamma_N\cap\Delta_N= \emptyset$ and $q_i(n)<q_{i+1}(n),\, i=1,...,\ell$ are nonnegative increasing functions taking on integer values on integers. It turns out that if $P\{\xi_0\in\Gamma_N\}$ and $P\{\xi_0\in\Delta_N\}$ decay in $N$ with the same speed then $\mathcal N_N$ 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 $\{ T^{q_i(n)}\omega\in A_n^b,\, i=1,...,\ell\}$ until the first occurence of another multiple recurrence $\{ T^{q_i(n)}\omega\in A_m^a,\, i=1,...,\ell\}$ where $A_m^a,\, A_n^b$ are cylinder sets of length $m$ and $n$ constructed by sequences $a,b\in\Omega$, 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".