Approximation of a sum of martingale differences generated by a bootstrap branching process

Abstract

Let \(\{Z(k), k\geq 0\}\) be a branching stochastic process with non-stationary immigration given by offspring distribution \(\{p_{j}(\theta),j\geq 0\}\) depending on unknown parameter \(\theta\in \Theta\). We estimate θ by an estimator \(\hat{\theta}_{n}\) based on sample \(\mathcal{X}_{n}=\{Z(i), i=1, {\ldots}, n\}\). Given \(\mathcal{X}_{n}\), we generate bootstrap branching process \(\{Z^{\mathcal{X}_{n}}(k), k\geq 0\}\) for each \( n=1, 2, {\ldots}\) with offspring distribution \(\{p_{j}(\hat{\theta}_{n}), j\geq 0\}\). In the paper we address the following question: How good must be estimator \(\hat{\theta}_{n}\), the bootstrap process to have the same asymptotic properties as the original process? We obtain conditions for the estimator which are sufficient and necessary for this in critical case. To derive these conditions we investigate a weighted sum of martingale differences generated by an array of branching processes. We provide a general functional limit theorem for this sum, which includes critical or nearly critical processes with increasing or stationary immigration and with large or fixed number of initial ancestors. It also includes processes without immigration with increasing random number of initial individuals. Possible applications in estimation theory of branching processes are also be provided.