A Berry-Esseen type inequality is proved via Stein's method for the logarithm of a Poisson randomly indexed branching process {Z Nt } , where {Z n } is a supercritical Galton-Watson process and {N t } is a Poisson process which is independent of {Z n } .

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Consider a Galton-Watson process {Z n } , the Lotka-Negaev estimator for offspring mean m is R n = Z n+1 /Z n. Let N t be a Poisson process independent of {Z n } , the continuous time process {Z Nt } is a Poisson randomly indexed branching process. We show the asymptotic distributions for {R t := R Nt }. Mathematics subject classification (2010): 60J80.

In a recent manuscript, Chu (2018) applied the self-normalized large deviations for i.i.d. random variables to the Lotka-Nagaev estimation of a supercritical Galton-Watson process. In this paper, we consider decay rates for the Lotka-Nagaev estimation of a supercritical branching process with immigration. We have two main contributions. On the one hand, Chu's paper considered the self-normalizing constants of second order, otherwise, we consider the maximum case. On the other hand, except for large deviations, we also studied the selfnormalized moderate deviations. The classical large deviation probabilities for Lotka-Negaev estimation show three different decay rates according to the degree of heavy tail of offspring distribution, but our results show that there is only one decay rate in the self-normalized version.Mathematics subject classification (2020): 60J80.