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 asymptotics 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.