A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights
Gerold Alsmeyer, Inst. Math. Statistics, Dept. Math. and Computer Science, University of Münster
Uwe Rösler, Math. Seminar, University of Kiel
Abstract
For real numbers $C,T_{1},T_{2},...$ we find all solutions
$mu$ to the
stochastic fixed point equation $Weqdistsum_{jge 1}T_{j}W_{j}+C$,
where $W,W_{1},W_{2},...$ are independent real-valued random variables with
distribution $mu$ and $eqdist$ means equality in distribution. All solutions
are
infinitely divisible. The set of solutions depends on the closed multiplicative
subgroup of ${Bbb R}_{*}={Bbb R}backslash{0}$ generated by the $T_{j}$.
If this group
is continuous, i.e. ${Bbb R}_{*}$ itself or the positive halfline ${BbbR}_{+}$,
then
all nontrivial fixed points are stable laws. In the remaining (discrete)
cases
further periodic solutions arise. A key observation is that the L'evy measure
of
any fixed point is harmonic with respect to $Lambda=sum_{jge 1}delta_{T_{j}}$,
i.e. $Gamma=GammastarLambda$, where $star$ means multiplicative convolution.
This will enable us to apply the powerful Choquet-Deny theorem.
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