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The empirical mean position of a branching Lévy process

Part of: Markov processes

Published online by Cambridge University Press:  23 November 2020

David Cheek  and
Seva Shneer
David Cheek*
Affiliation:
Harvard University
Seva Shneer*
Affiliation:
Harvard University
*
*Postal address: Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts, USA. Email address: dmcheek@g.harvard.edu
**Postal address: School of MACS, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: v.shneer@hw.ac.uk
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Abstract

We consider a supercritical branching Lévy process on the real line. Under mild moment assumptions on the number of offspring and their displacements, we prove a second-order limit theorem on the empirical mean position.

Keywords

Branching Lévy processbranching random walkbranching Brownian motionempirical mean positionempirical mean depth

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1252 - 1259
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Copyright
© Applied Probability Trust 2020

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References

Aldous, D. J. (2001). Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today. Statist. Sci. 16, 23–34.10.1214/ss/998929474CrossRefGoogle Scholar
Athreya, K. and Ney, P. (1972). Branching Processes. Springer.10.1007/978-3-642-65371-1CrossRefGoogle Scholar
Biggins, J. (1990). The central limit theorem for the supercritical branching random walk, and related results. Stoch. Process. Appl. 34, 255–274.10.1016/0304-4149(90)90018-NCrossRefGoogle Scholar
Biggins, J. (1995). The growth and spread of the general branching random walk. Ann. Appl. Prob. 5, 1008–1024.10.1214/aoap/1177004604CrossRefGoogle Scholar
Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31, 531–581.10.1002/cpa.3160310502CrossRefGoogle Scholar
Chauvin, B., Klein, T., Marckert, J.-F. and Rouault, A. (2005). Martingales and profile of binary search trees. Electron. J. Prob. 10, 420–435.10.1214/EJP.v10-257CrossRefGoogle Scholar
Duffy, K. R., Meli, G. and Shneer, S. (2019). The variance of the average depth of a pure birth process converges to 7. Statist. Prob. Lett. 150, 88–93.10.1016/j.spl.2019.02.015CrossRefGoogle Scholar
Durrett, R. (2013). Population genetics of neutral mutations in exponentially growing cancer cell populations. Ann. Appl. Prob. 23, 230–250.10.1214/11-AAP824CrossRefGoogle ScholarPubMed
Felsenstein, J. (2004). Inferring Phylogenies. Sinauer Associates.Google Scholar
Gantert, N. and Höfelsauer, T. (2018). Large deviations for the maximum of a branching random walk. Electron. Commun. Prob. 23, 34.10.1214/18-ECP135CrossRefGoogle Scholar
Gao, Z. and Liu, Q. (2018). Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time. Bernoulli 24, 772–800.10.3150/16-BEJ895CrossRefGoogle Scholar
Louidor, O. and Tsairi, E. (2017). Large deviations for the empirical distribution in the general branching random walk. Available at .Google Scholar
Meli, G., Weber, T. S. and Duffy, K. R. (2019). Sample path properties of the average generation of a Bellman–Harris process. J. Math. Biol. 79, 673–704.10.1007/s00285-019-01373-0CrossRefGoogle ScholarPubMed