Electronic Communications in Probability

A strong law of large numbers for branching processes: almost sure spine events

Simon Harris and Matthew Roberts

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We demonstrate a novel strong law of large numbers for branching processes, with a simple proof via measure-theoretic manipulations and spine theory. Roughly speaking, any sequence of events that eventually occurs almost surely for the spine entails the almost sure convergence of a certain sum over particles in the population.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 28, 6 pp.

Accepted: 8 May 2014
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

branching processes spines martingales strong law

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Harris, Simon; Roberts, Matthew. A strong law of large numbers for branching processes: almost sure spine events. Electron. Commun. Probab. 19 (2014), paper no. 28, 6 pp. doi:10.1214/ECP.v19-2641. https://projecteuclid.org/euclid.ecp/1465316730

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  • Hardy, Robert; Harris, Simon C. A spine approach to branching diffusions with applications to $\scr L^ p$-convergence of martingales. SÄ‚Å minaire de ProbabilitÄ‚Å s XLII, 281–330, Lecture Notes in Math., 1979, Springer, Berlin, 2009.
  • S.C. Harris, M. Hesse, and A.E. Kyprianou. Branching brownian motion in a strip: survival near criticality. 2012. Preprint: arXiv:1212.1444v1.
  • M.I. Roberts. Spine changes of measure and branching diffusions. PhD thesis, University of Bath, 2010. Available online: http://people.bath.ac.uk/mir20/thesis.pdf.