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The Compensation Approach for Walks With Small Steps in the Quarter Plane

Published online by Cambridge University Press:  11 January 2013

IVO J. B. F. ADAN ,
JOHAN S. H. van LEEUWAARDEN  and
KILIAN RASCHEL
IVO J. B. F. ADAN
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: i.j.b.f.adan@tue.nl, j.s.h.v.leeuwaarden@tue.nl)
JOHAN S. H. van LEEUWAARDEN
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: i.j.b.f.adan@tue.nl, j.s.h.v.leeuwaarden@tue.nl)
KILIAN RASCHEL
Affiliation:
CNRS and Université de Tours, Faculté des Sciences et Techniques, Parc de Grandmont, 37200 Tours, France (e-mail: kilian.raschel@lmpt.univ-tours.fr)
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Abstract

This paper is the first application of the compensation approach (a well-established theory in probability theory) to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane +2 with a step set that is a subset of

\[ \{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}\]
in the interior of +2. We derive an explicit expression for the generating function which turns out to be non-holonomic, and which can be used to obtain exact and asymptotic expressions for the counting numbers.

Keywords

Primary 05A15Secondary 05A16
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 161 - 183
Copyright
Copyright © Cambridge University Press 2013

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