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Peaks and Eulerian numbers in a random sequence

Part of: Distribution theory Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Di Warren  and
E. Seneta
Di Warren*
Affiliation:
University of Sydney
E. Seneta*
Affiliation:
University of Sydney
*
∗Postal address for both authors: School of Mathematics and Statistics, F07, University of Sydney, N.S.W. 2006, Australia.
∗Postal address for both authors: School of Mathematics and Statistics, F07, University of Sydney, N.S.W. 2006, Australia.
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Abstract

We consider the exact distribution of the number of peaks in a random permutation of the integers 1, 2, ···, n. This arises from a test of whether n successive observations from a continuous distribution are i.i.d. The Eulerian numbers, which figure in the p.g.f., are then shown to provide a link between the simpler problem of ascents (which has been thoroughly analysed) and both our problem of peaks and similar problems on the circle. This link then permits easy deduction of certain general properties, such as linearity in n of the cumulants, in the more complex settings. Since the focus of the paper is on exact distributional results, a uniform bound on the deviation from the limiting normal is included. A secondary purpose of the paper is synthesis, beginning with the more familiar setting of peaks and troughs.

Keywords

ASCENTSCENTRAL LIMIT PROPERTYCIRCLECONVERGENCE RATECUMULANTSEULERIAN NUMBERSm-DEPENDENCEPEAKS AND TROUGHSPERMUTATIONPHASES

MSC classification

Primary: 60G10: Stationary processes
Secondary: 62E10: Characterization and structure theory 62G10: Hypothesis testing
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 101 - 114
Copyright
Copyright © Applied Probability Trust 1996 

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