Local large deviations: McMillian theorem for multitype Galton-Watson process

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Abstract

In this article, we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential UK(⋅, π) for the Galton-Watson process, where π is the normalized eigenvector corresponding to the leading Perron-Frobenius eigenvalue 1 of the transition matrix A(⋅, ⋅) defined from K, the transition kernel. We show that the Kullback action or the deviation function, J (π, ρ), with respect to an empirical offspring measure, ρ, is the Legendre dual of UK(⋅, π). From the LLDP, we deduce a conditional large deviation principle and a weak variant of the classical McMillian theorem for the multitype Galton-Watson process. To be specific, given any empirical offspring measure ϖ, we show that the number of critical multitype Galton-Watson processes on n vertices is approximately (Formula presented), where Hϖ is a suitably defined entropy.

Original languageEnglish
Pages (from-to)2307-2319
Number of pages13
JournalFar East Journal of Mathematical Sciences
Volume102
Issue number10
DOIs
Publication statusPublished - Nov 2017

Keywords

  • Eigenvector
  • Galton-Watson process
  • Kullback action
  • Local large deviation
  • Perron-Frobenius eigenvalue
  • Spectral potential
  • Typed trees
  • Variational principle

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