Electronic Journal of Probability

A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights

Gerold Alsmeyer and Uwe Rösler

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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.

Article information

Electron. J. Probab., Volume 11 (2006), paper no. 2, 27-56.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60E10: Characteristic functions; other transforms 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Stochastic fixed point equation weighted branching process infinite divisibility L'evy measure Choquet-Deny theorem stable distribution

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Alsmeyer, Gerold; Rösler, Uwe. A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights. Electron. J. Probab. 11 (2006), paper no. 2, 27--56. doi:10.1214/EJP.v11-296. https://projecteuclid.org/euclid.ejp/1464730537

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