Validity of the bootstrap in the critical process with a non-stationary immigration

In the critical branching process with a stationary immigration, the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance (n) and (n) are finite for each n epsilon 1 and are regularly varying sequences with non-negative exponents and , respectively. We prove that if (n) and (n)=o(n2(n)) as n, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least-squares estimator in the sense of convergence in probability. Monte Carlo and bootstrap simulations for the process confirm the theoretical findings in the paper and highlight the validity and utility of the bootstrap as it mimics the Monte Carlo pivots even when generation size is small.