A Phase Transition for Large Values of Bifurcating Autoregressive Models

We describe the asymptotic behavior of the number Z(n)[a(n), infinity) of individuals with a large value in a stable bifurcating autoregressive process, where a(n) -> infinity. The study of the associated first moment is equivalent to the annealed large deviation problem of an autoregressive process in a random environment. The trajectorial behavior of Z(n)[a(n), infinity) is obtained by the study of the ancestral paths corresponding to the large deviation event together with the environment of the process. This study of large deviations of autoregressive processes in random environment is of independent interest and achieved first. The estimates for bifurcating autoregressive process involve then a law of large numbers for non-homogenous trees. Two regimes appear in the stable case, depending on whether one of the autoregressive parameters is greater than 1 or not. It yields different asymptotic behaviors for large local densities and maximal value of the bifurcating autoregressive process.