Fixed Points of the Smoothing Transform: the Boundary Case
John D Biggins, University of Sheffield
Andreas E Kyprianou, Heriot-Watt University
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 $Eleft[sum A_{i} log A_i right]<0$.
New results on the asymptotic behaviour of the solutions near zero
in the boundary case, where
$Eleft[sum A_{i} log A_i right]=0$, are obtained.
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