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On degenerate sums of $$m$$-dependent variables. (English) Zbl 1334.60021
Summary: 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 $$\operatorname{var}(X_{i})\neq 0$$. 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 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.

##### MSC:
 60F05 Central limit and other weak theorems 60G10 Stationary stochastic processes 60C05 Combinatorial probability
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##### References:
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