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 language | English |
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Pages (from-to) | 2307-2319 |
Number of pages | 13 |
Journal | Far East Journal of Mathematical Sciences |
Volume | 102 |
Issue number | 10 |
DOIs | |
Publication status | Published - Nov 2017 |
Keywords
- Eigenvector
- Galton-Watson process
- Kullback action
- Local large deviation
- Perron-Frobenius eigenvalue
- Spectral potential
- Typed trees
- Variational principle