Journal of Applied Probability

On degenerate sums of m-dependent variables

Svante Janson

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It is well known that the central limit theorem holds for partial sums of a stationary sequence (Xi) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(Xi) ≠ 0. We show that this happens only in the case when Xi - EXi = Yi - Yi-1 for an (m - 1)-dependent stationary sequence (Yi) 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.

Article information

J. Appl. Probab., Volume 52, Number 4 (2015), 1146-1155.

First available in Project Euclid: 22 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes
Secondary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability

m-dependent stationary sequence block factor random tree


Janson, Svante. On degenerate sums of m -dependent variables. J. Appl. Probab. 52 (2015), no. 4, 1146--1155. doi:10.1239/jap/1450802758.

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