On the statistics of branching processes

Let $\mu_t(n)$, $t=0,1,2,\dots,$ be a Galton–Watson process, starting from $n$ particles. We show that when $n,t\to\infty$ the estimator
$$ \widehat A_t(n)=\frac{\sum_{k=1}^t\mu_k(n)}{\sum_{k=0}^{t-1}\mu_k(n)} $$
for the expectation $A=\mathbf E\mu_1(1)$ is consistent and asymptoticaly unbiased. We obtain limit distributions for $\widehat A_t(n)$ in the subcritical, critical and supercritical cases.