Limit Laws of Estimators for Critical Multi - Type Galton – Watson Processes

We consider the asymptotics of various estimators based on a large sample of branching trees from a critical multi-type Galton– Watson process, as the sample size increases to infinity. The asymp-totics of additive functions of trees, such as sizes of trees and frequencies of types within trees, a higher-order asymptotic of the " relative frequency " estimator of the left eigenvector of the mean matrix, a higher-order joint asymptotic of the maximum likelihood estimators of the offspring probabilities and the consistency of an estimator of the right eigenvector of the mean matrix, are established. 1. Introduction. This article considers the asymptotics of estimators associated with critical multi-type Galton–Watson (GW) processes. A GW process is called critical if the largest eigenvalue of its mean matrix is 1 (see below for details). For such a process, a branching tree is finite with probability 1, but the expectation of its size is infinite. The estimators considered here are based on a large sample of terminating branching trees, and the asymptotics refer to the probabilistic behavior as the sample size n → ∞. The study on large sample asymptotics of parameter estimators for simple (i.e., single type) GW processes has a quite long history (cf. [20]). The idea of using increasingly large sample of individual trees for estimation dates from as early as [24], and much progress has been made since then (cf. [7, 8] and references therein). This setting of parameter estimation is widely used in computational linguistics [5, 16], where large samples of tree-structured parses of sentences are available. On the general issues of parameter estimation or asymptotics related to simple or multi-type GW processes, there is now extensive literature available (e.