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


 
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1. Title Title of document A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights
 
2. Creator Author's name, affiliation, country Gerold Alsmeyer; Inst. Math. Statistics, Dept. Math. and Computer Science, University of Münster
 
2. Creator Author's name, affiliation, country Uwe Rösler; Math. Seminar, University of Kiel
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Stochastic fixed point equation; weighted branching process; infinite divisibility; L'evy measure; Choquet-Deny theorem; stable distribution
 
3. Subject Subject classification 60E07; 60E10; 60J80
 
4. Description 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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7. Date (YYYY-MM-DD) 2006-01-26
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/296
 
10. Identifier Digital Object Identifier 10.1214/EJP.v11-296
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 11
 
12. Language English=en
 
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