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On degenerate sums of m-dependent variables

Part of: Combinatorial probability Stochastic processes Limit theorems

Published online by Cambridge University Press:  30 March 2016

Svante Janson
Svante Janson*
Affiliation:
Uppsala University
*
∗Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. Email address: svante@math.uu.se
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Abstract

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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.

Keywords

m-dependentstationary sequenceblock factorrandom tree

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1146 - 1155
Copyright
Copyright © Applied Probability Trust 2015 

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