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.
"On degenerate sums of m-dependent variables." J. Appl. Probab. 52 (4) 1146 - 1155, December 2015. https://doi.org/10.1239/jap/1450802758