A characterization of $\boldsymbol{m}$-dependent stationary infinitely divisible sequences with applications to weak convergence

Abstract

$m$-dependent stationary infinitely divisible sequences are characterized as a class of generalized finite moving average sequences via the structure of the associated Lévy measure. This characterization is used to find necessary and sufficient conditions for the weak convergence of centered and normalized partial sums of $m$-dependent stationary infinitely divisible sequences. Partial sum convergence for stationary infinitely divisible sequences that can be approximated by $m$-dependent ones is then studied.