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

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 eigen vector corresponding to the leading Perron-Frobenius eigen value 1l 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 ̟, we show that the number of critical multitype Galton-Watson processes on n vertices is approximately en〈H̟ , π〉, where H̟ is a suitably defined entropy.