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 (X_i) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(X_i) ≠ 0. We show that this happens only in the case when X_i - EX_i = Y_i - Y_i-1 for an (m - 1)-dependent stationary sequence (Y_i) 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.