Local Large Deviations: McMillian Theorem for multitype Galton-Watson Processes

TL;DR

This paper establishes a local large deviation principle for critical multitype Galton-Watson processes using spectral potential theory, leading to a variant of McMillian's theorem and explicit asymptotic counts.

Contribution

It introduces a spectral potential framework and derives a local large deviation principle for multitype Galton-Watson processes, extending classical results.

Findings

Proves LLDP for critical multitype Galton-Watson processes.

Derives a conditional large deviation principle and a McMillian-type theorem.

Provides asymptotic enumeration of processes based on empirical offspring measures.

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 $U_{\skrik}(\,\cdot,\,\pi)$ for the Galton-Watson process, where $\pi$ is the normalized eigen vector corresponding to the leading \emph{Perron-Frobenius eigen value } $\1$ of the transition matrix $\skria(\cdot,\,\cdot)$ defined from ${\skrik},$ the transition kernel. We show that the Kullback action or the deviation function, $J(\pi,\rho),$ with respect to an empirical offspring measure, $\rho,$ is the Legendre dual of $U_{\skrik}(\,\cdot,\,\pi).$ 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 $\varpi,$ we show that the number of critical multitype Galton-Watson processes on $n$ vertices is approximately $e^{n\langle \skrih_{\varpi},\,\pi\rangle},$ where $\skrih_{\varpi}$ is a suitably defined entropy.