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A stochastic fixed point equation related to weighted branching with deterministic weights. (English) Zbl 1110.60080

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

MSC:

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