On degenerate sums of m-dependent variables

Abstract

It is well known that the central limit theorem holds for partial sums of a stationary sequence (Xi ) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(Xi ) ≠ 0. We show that this happens only in the case when Xi – EXi = Yi – Yi –1 for an (m − 1)-dependent stationary sequence (Yi ) with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to be degenerate. Two applications to subtree counts in random trees are given.