The growth of iterates of multivariate generating functions

by J.D.Biggins

2000 Mathematics Subject Classification: Primary 39B12; Secondary 05A16, 60J80, 60F10

Abstract

The vector-valued function $m(\theta)$ of a $p$-vector $\theta$ has components $m_1(\theta)$, $m_2(\theta)$, \ldots, $m_p(\theta)$. For each $i$, $\exp(m_i(-\theta))$ is the (multivariate) Laplace transform of a discrete measure, $\mu_i$, 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 allow certain results in Jones (2004) on multi\-type branching to be proved under significantly weaker conditions.

Availability:

Preprint 566/06, Department of Probability and Statistics, University of Sheffield. (August 2006).

Very minor revision, and reformating, pdf file

Accepted (August 22, 2006). Trans. Amer. Math. Soc. 360 (2008), 4305-4334

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