Large Deviation Results for Critical Multitype Galton-Watson Trees

Abstract

In this article, we prove a joint large deviation principle in n for the empirical pair measure and empirical offspring measure of critical multitype Galton-Watson trees conditioned to have exactly n vertices in the weak topology. From this result we extend the large deviation principle for the empirical pair measures of Markov chains on simply generated trees to cover offspring laws which are not treated by [DMS03, Theorem 2.1]. For the case where the offspring law of the tree is a geometric distribution with parameter 1 2 , we get an exact rate function. All our rate functions are expressed in terms of relative entropies.

LOCAL LARGE DEVIATIONS: McMILLIAN THEOREM FOR MULTITYPE GALTON-WATSON PROCESS

“…The main technique used is exponential change of measure. These results have thrown more insight on the results of [3] and [4].…”

“…Next, we state a key ingredient (Lemma 3.1) in the proof of our main result, Theorem 2.1. This Lemma gives remarkable properties of 2.2 above, which will help us circumvent the topological problems faced in [4] and [5]. To state the lemma, we denote by C the space of continuous functions g : Y × Y * → R and notice that, the proof below follows similar ideas as the proof of [1, Lemma 2.2] for the empirical measures on measurable spaces.…”

“…Recently, Doku-Amponsah [2] found a Lossy version of the result [2, Theorem 2.1]. The article [4] extended the LDP result of [5,Theorem 2.2] for the multitype Galton-Watson Process to cover offspring laws (e.g. geometric 1 2 ), which were not covered by [5], in the weak topology.…”

“…Presently there exists some Large deviation Principle(LDP) and Basic Information Theory for the multitype Galton-Watson Process.See, example Dembo et al [5], Doku-Amponsah [4], Doku-Amponsah [3], Doku-Amponsah [2]. In [5], LDPs were proved for empirical measures of the multitype Galton-Watson trees with the exponential moments of offspring transition kernel being finite, in a topology stronger than weak topology.…”

“…To be specific about this approach, we define a spectral potential of the multitype Galton-Watson trees and use it to define an extension of the relative entropy, which we also call the Kullback action. The Kullback action has the rate function of the LDPs in [4] and [5] as special cases. By a proper exponential change of measure,using the Kullback action we prove the LLDP, and from the LLDP we obtain the classical McMillan-Breiman Theorem as a particular case.…”