Electronic Journal of Probability
- Electron. J. Probab.
- Volume 7 (2002), paper no. 8, 25 pp.
Asymptotics of Certain Coagulation-Fragmentation Processes and Invariant Poisson-Dirichlet Measures
We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $\beta_m$ (if the sampled parts are distinct) or splitting the part with probability $\beta_s$, according to a law $\sigma$ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $\sigma$ is the uniform measure, then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of "analytic" invariant measures. We also derive transience and recurrence criteria for these chains.
Electron. J. Probab., Volume 7 (2002), paper no. 8, 25 pp.
Accepted: 14 February 2002
First available in Project Euclid: 16 May 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
This work is licensed under aCreative Commons Attribution 3.0 License.
Mayer-Wolf, Eddy; Zeitouni, Ofer; Zerner, Martin. Asymptotics of Certain Coagulation-Fragmentation Processes and Invariant Poisson-Dirichlet Measures. Electron. J. Probab. 7 (2002), paper no. 8, 25 pp. doi:10.1214/EJP.v7-107. https://projecteuclid.org/euclid.ejp/1463434881