# 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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Other publications by J.D.
Biggins

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