Asymptotic behaviour of a bootstrap branching process

In this paper we study weak convergence of sequences of random probability measures generated by bootstrap branching processes. Let $\{Z(k),k\ge0\}$ be a branching stochastic process with non-stationary immigration given by an offspring distribution $\{p_j(\theta),j\ge0\}$ depending on the unknown parameter $\theta\in\Theta\subseteq\mathbb R$. We estimate $\theta$ by an estimator $\hat\theta_n$ based on a sample $\{Z(i)$, $i=1,\dots,n\}$. Given $\hat\theta_n$, we generate the bootstrap branching process (BBP) $\{Z^{\hat\theta_n}(k),k\ge0\}$ for each $n=1,2,\dots$ with the offspring distribution $\{p_j(\hat\theta_n),j\ge0\}$. We derive conditions on the estimator $\hat\theta_n$ which are sufficient and necessary for the bootstrap process to have the same asymptotic properties as the original process. These results allow us to investigate the validity of the bootstrap without using an explicit form of the estimator. In applications of branching processes obtaining samples of large sizes is difficult. Therefore, the bootstrap process can be used to generate multiple samples of large size.