We describe the asymptotic behavior of the conditional least squares estimator of the offspring mean for subcritical strongly stationary Galton--Watson processes with regularly varying immigration with tail index $\alpha \in (1,2)$. The limit law is the ratio of two dependent stable random variables with indices $\alpha/2$ and $2\alpha/3$, respectively, and it has a continuously differentiable density function. We use point process technique in the proofs.

中文翻译：

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具有规律变化移民的亚临界强平稳 Galton-Watson 过程的统计推断

我们描述了子临界强平稳 Galton-Watson 过程的后代均值的条件最小二乘估计量的渐近行为，该过程具有规则变化的移民，尾指数为 $\alpha\in (1,2)$。极限定律是指数分别为$\alpha/2$和$2\alpha/3$的两个相依稳定随机变量的比值，并且具有连续可微的密度函数。我们在证明中使用点处理技术。