We consider an observed subcritical Galton Watson process $\{Y_n,\ n\in \mathbb{Z} \}$ with correlated stationary immigration process $\{\epsilon_n,\ n\in \mathbb{Z} \}$. Two situations are presented. The first one is when $\mbox{Cov}(\epsilon_0,\epsilon_k)=0$ for $k$ larger than some $k_0$: a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when $\{\epsilon_n,\ n\in \mathbb{Z} \}$ has general correlation structure: under mixing assumptions, we exhibit an estimator for the the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.

翻译：本文考虑一个观测到的次临界伽尔顿-沃森过程 $\{Y_n,\ n\in \mathbb{Z} \}$，其伴随有相关平稳移民过程 $\{\epsilon_n,\ n\in \mathbb{Z} \}$。我们提出两种情形。第一种情形是当 $\mbox{Cov}(\epsilon_0,\epsilon_k)=0$ 对于大于某个 $k_0$ 的 $k$ 成立时：给出了繁殖率和平均移民率的一致估计量，并证明了中心极限定理。第二种情形是当 $\{\epsilon_n,\ n\in \mathbb{Z} \}$ 具有一般相关结构时：在混合假设下，我们构造了繁殖率对数的估计量，并证明其以显式速度均方收敛。此外，当混合系数足够快衰减时，我们给出了该估计量的两项展开式并予以证明。文中还提供了数值算例。