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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.

60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes
60C05 Combinatorial probability
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