Limiting mixture distributions for AR(1) model indexed by a branching process

Abstract

First order autoregressive model indexed by a supercritical Galton–Watson branching process is discussed. Limiting distributions of the least squares estimates are derived both for the stationary and explosive cases. It is shown that a certain random variable inherent in the branching process is acting as a mixing variable in limiting mixture distributions. In particular, with explosive Gaussian case, we obtain a mixture of Cauchy distributions rather than Cauchy.