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

Simon C Harris (University of Bath)
Matthew I Roberts (University of Warwick)

Abstract


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.


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Pages: 1-6

Publication Date: May 8, 2014

DOI: 10.1214/ECP.v19-2641

References

  • Hardy, Robert; Harris, Simon C. A spine approach to branching diffusions with applications to $\scr L^ p$-convergence of martingales. Séminaire de Probabilités XLII, 281--330, Lecture Notes in Math., 1979, Springer, Berlin, 2009. MR2599214
  • 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.
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