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On degenerate sums of m-dependent variables

Published online by Cambridge University Press:  30 March 2016

Svante Janson*
Affiliation:
Uppsala University
*
Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. Email address: [email protected]
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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 = YiYi–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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2015 

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