# The smoothing transform: the boundary case

by J.D.Biggins and A.E Kyprinaou

2000 Mathematics Subject Classification: 60J80

## Abstract

Let $A=(A_1,A_2,A_3,\ldots)$ be a random sequence of
non-negative numbers
that are ultimately zero with $E[\sum A_i]=1$ and $E
\left[\sum A_{i} \log A_i \right] \leq 0$.
The uniqueness of the
non-negative fixed points of the associated smoothing transform is
considered. These fixed points
are solutions to the functional equation $\Phi(\psi)= E
\left[ \prod_{i} \Phi(\psi A_i) \right], $ where $\Phi$ is the Laplace
transform of a non-negative random variable. The study complements,
and extends, existing
results on the case when $E\left[\sum A_{i} \log A_i \right]<0$.
New results on the asymptotic behaviour
of the solutions
near zero in the boundary case, where
$E\left[\sum A_{i} \log A_i \right]=0$, are obtained.

## Availability

Preprint 516/01,
Department of Probability and Statistics, University of Sheffield.
(December 2001)
pdf version

pdf version 2

Electronic Journal of Probability (2005) **10**,
609-631

Other publications by J.D. Biggins

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