Abstract
For a
$\psi $
-mixing process
$\xi _0,\xi _1,\xi _2,\ldots $
we consider the number
${\mathcal N}_N$
of multiple returns
$\{\xi _{q_{i,N}(n)}\in {\Gamma }_N,\, i=1,\ldots ,\ell \}$
to a set
${\Gamma }_N$
for n until either a fixed number N or until the moment
$\tau _N$
when another multiple return
$\{\xi _{q_{i,N}(n)}\in {\Delta }_N,\, i=1,\ldots ,\ell \}$
, takes place for the first time where
${\Gamma }_N\cap {\Delta }_N=\emptyset $
and
$q_{i,N}$
,
$i=1,\ldots ,\ell $
are certain functions of n taking on non-negative integer values when n runs from 0 to N. The dependence of
$q_{i,N}(n)$
on both n and N is the main novelty of the paper. Under some restrictions on the functions
$q_{i,N}$
we obtain Poisson distributions limits of
${\mathcal N}_N$
when counting is until N as
$N\to \infty $
and geometric distributions limits when counting is until
$\tau _N$
as
$N\to \infty $
. We obtain also similar results in the dynamical systems setup considering a
$\psi $
-mixing shift T on a sequence space
${\Omega }$
and studying the number of multiple returns
$\{ T^{q_{i,N}(n)}{\omega }\in A^a_n,\, i=1,\ldots ,\ell \}$
until the first occurrence of another multiple return
$\{ T^{q_{i,N}(n)}{\omega }\in A^b_m,\, i=1,\ldots ,\ell \}$
where
$A^a_n,\, A_m^b$
are cylinder sets of length n and m constructed by sequences
$a,b\in {\Omega }$
, respectively, and chosen so that their probabilities have the same order.
Limit theorems for numbers of multiple returns in non-conventional arrays
