Asymptotic distributions for weighted estimators of the offspring mean in a branching process

It is known that conditional least squares estimator (CLSE) of the offspring mean for the process with a stationary immigration is not asymptotically normal. In the paper, we demonstrate that for the process with non-stationary immigration it may have a normal limit distribution. Considering a discrete time branching process Z(n) with time-dependent immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively, we show that 1 + 2α is the threshold for asymptotic normality of the estimator. It will be proved that if β < 1 + 2α, the estimator is asymptotically normal with two different normalizing factors, and if β > 1 + 2α its limiting distribution is not normal, but can be expressed in terms of certain functionals of the time-changed Wiener process. When β = 1 + 2α, the limiting distribution depends on the behavior of the slowly varying parts of the mean and variance. We derive all possible limit distributions of the weighted CLSE based on observations {Z(r+1),Z(r+2),...,Z(n)} as n → ∞ and r=[nε], 0≤ ε < 1. Conditions guaranteeing the strong consistency of the proposed estimator will be derived.