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Concave Majorants of Random Walks and Related Poisson Processes

Published online by Cambridge University Press:  18 August 2011

JOSH ABRAMSON  and
JIM PITMAN
JOSH ABRAMSON
Affiliation:
Department of Statistics, University of California at Berkeley, CA 94720, USA (e-mail: josh@stat.berkeley.edu, pitman@stat.berkeley.edu)
JIM PITMAN
Affiliation:
Department of Statistics, University of California at Berkeley, CA 94720, USA (e-mail: josh@stat.berkeley.edu, pitman@stat.berkeley.edu)
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Abstract

We offer a unified approach to the theory of concave majorants of random walks, by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant of a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 651 - 682
Copyright
Copyright © Cambridge University Press 2011

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