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It is well-known that the central limit theorem holds for partial sums of astationary sequence $(X_i)$ of $m$-dependent random variables with finitevariance; however, the limit may be degenerate with variance 0 even if$\mathrm{Var}(X_i)\neq0$. We show that this happens only in the case when$X_i-\mathbb E X_i=Y_i-Y_{i-1}$ for an $(m-1)$-dependent stationary sequence$(Y_i)$ with finite variance (a result implicit in earlier results), and give aversion for block factors. This yields a simple criterion that is a sufficientcondition for the limit not to degenerate. Two applications to subtree countsin random trees are given.

On degenerate sums of $m$-dependent variables