How to cite item

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

   
@article{EJP296,
	author = {Gerold Alsmeyer and Uwe Rösler},
	title = {A Stochastic Fixed Point Equation Related to Weighted Branching with Deterministic Weights},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {11},
	year = {2006},
	keywords = {Stochastic fixed point equation; weighted branching process; infinite divisibility; L'evy measure; Choquet-Deny theorem; stable distribution},
	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.},
	pages = {no. 2, 27-56},
	issn = {1083-6489},
	doi = {10.1214/EJP.v11-296},    
        url = {http://ejp.ejpecp.org/article/view/296}}