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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The growth of iterates of multivariate generating functions


Author: J. D. Biggins
Journal: Trans. Amer. Math. Soc. 360 (2008), 4305-4334
MSC (2000): Primary 39B12; Secondary 05A16, 60J80, 60F10
Published electronically: March 14, 2008
MathSciNet review: 2395174
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Abstract | References | Similar Articles | Additional Information

Abstract: The vector-valued function $ m(\theta)$ of a $ p$-vector $ \theta$ has components $ m_1(\theta), m_2(\theta), \dots, m_p(\theta)$. For each $ i$, $ \exp(m_i(-\theta))$ is the (multivariate) Laplace transform of a discrete measure concentrated on $ [0,\infty)^p$ with only a finite number of atoms. The main objective is to give conditions for the functional iterates $ m^{(n)} $ of $ m$ to grow like $ \rho^n$ for a suitable $ \rho>1$. The initial stimulus was provided by results of Miller and O'Sullivan (1992) on enumeration issues in `context free languages', results which can be improved using the theory developed here. The theory also allows certain results in Jones (2004) on multitype branching to be proved under significantly weaker conditions.


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Additional Information

J. D. Biggins
Affiliation: Department of Probability and Statistics, The University of Sheffield, Sheffield, S3 7RH, United Kingdom
Email: J.Biggins@sheffield.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04408-5
PII: S 0002-9947(08)04408-5
Keywords: Generating functions, iterates, maximum growth rate, non-negative matrices, functional iteration, multitype branching processes, enumeration
Received by editor(s): December 23, 2005
Received by editor(s) in revised form: August 17, 2006
Published electronically: March 14, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.