It is shown that certain estimators of the offspring and immigration means in a subcritical simple branching process with immigration are strongly consistent and obey the central limit theorem and law of the iterated logarithm under natural conditions.
"Asymptotic Results for Estimators in a Subcritical Branching Process with Immigration." Ann. Probab. 4 (2) 319 - 325, April, 1976. https://doi.org/10.1214/aop/1176996137