Electronic Journal of Probability

On the Speed of Coming Down from Infinity for $\Xi$-Coalescent Processes

Vlada Limic

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Abstract

The $\Xi$-coalescent processes were initially studied by Möhle and Sagitov (2001), and introduced by Schweinsberg (2000) in their full generality. They arise in the mathematical population genetics as the complete class of scaling limits for genealogies of Cannings' models. The $\Xi$-coalescents generalize $\Lambda$-coalescents, where now simultaneous multiple collisions of blocks are possible. The standard version starts with infinitely many blocks at time $0$, and it is said to come down from infinity if its number of blocks becomes immediately finite, almost surely. This work builds on the technique introduced recently by Berstycki, Berestycki and Limic (2009), and exhibits deterministic ``speed'' function - an almost sure small time asymptotic to the number of blocks process, for a large class of $\Xi$-coalescents that come down from infinity.

Article information

Source
Electron. J. Probab. Volume 15 (2010), paper no. 8, 217-240.

Dates
Accepted: 1 March 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819793

Digital Object Identifier
doi:10.1214/EJP.v15-742

Mathematical Reviews number (MathSciNet)
MR2594877

Zentralblatt MATH identifier
1203.60111

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60F99: None of the above, but in this section 92D25: Population dynamics (general)

Keywords
Exchangeable coalescents small-time asymptotics coming down from infinity martingale technique

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Limic, Vlada. On the Speed of Coming Down from Infinity for $\Xi$-Coalescent Processes. Electron. J. Probab. 15 (2010), paper no. 8, 217--240. doi:10.1214/EJP.v15-742. https://projecteuclid.org/euclid.ejp/1464819793


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