Asymptotic distribution of the CLSE in a critical process with immigration

It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n^2/3$n^{2/3}$. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents α$\alpha$ and β$\beta$, respectively. We prove that if β<1+2α$\beta <1+2\alpha$, the CLSE is asymptotically normal with two different normalization factors and if β>1+2α$\beta >1+2\alpha$, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When β=1+2α$\beta =1+2\alpha$ the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.