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2006 A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights
Gerold Alsmeyer, Uwe Rösler
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Electron. J. Probab. 11: 27-56 (2006). DOI: 10.1214/EJP.v11-296

Abstract

For real numbers $C,T_{1},T_{2},...$ we find all solutions $\mu$ to the stochastic fixed point equation $W \sim\sum_{j\ge 1}T_{j}W_{j}+C$, where $W,W_{1},W_{2},...$ are independent real-valued random variables with distribution $\mu$ and $\sim$ means equality in distribution. All solutions are infinitely divisible. The set of solutions depends on the closed multiplicative subgroup of ${ R}_{*}={ R}\backslash\{0\}$ generated by the $T_{j}$. If this group is continuous, i.e. ${R}_{*}$ itself or the positive halfline ${R}_{+}$, then all nontrivial fixed points are stable laws. In the remaining (discrete) cases further periodic solutions arise. A key observation is that the Levy measure of any fixed point is harmonic with respect to $\Lambda=\sum_{j\ge 1}\delta_{T_{j}}$, i.e. $\Gamma=\Gamma\star\Lambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet-Deny theorem.

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Gerold Alsmeyer. Uwe Rösler. "A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights." Electron. J. Probab. 11 27 - 56, 2006. https://doi.org/10.1214/EJP.v11-296

Information

Accepted: 26 January 2006; Published: 2006
First available in Project Euclid: 31 May 2016

zbMATH: 1110.60080
MathSciNet: MR2199054
Digital Object Identifier: 10.1214/EJP.v11-296

Subjects:
Primary: 60E07
Secondary: 60E10, 60J80

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