Estimation of the mean for critical branching process and its bootstrap approximation

This article focuses attention on the estimation of the mean of a sequence of branching process with immigration in the critical case. We get the limiting distribution of the pivot, and adopt a bootstrap procedure to bootstrap the least-square estimator with bootstrap sample size $m$ less than the size $n$ of the original sample. Under the assumptions that $m\rightarrow \infty $ and $m/n\rightarrow 0$ , the convergence in probability of the bootstrap distribution function is also established.