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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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