### Large deviations for Lotka-Nagaev estimator of a randomly indexed branching process

#### Abstract

Consider a continuous time process $\{Y_t=Z_{N_t},t\geq 0\}$, where $\{Z_n\}$ is a supercritical Galton--Watson process and $\{N_t\}$ is a renewal process which is independent of $\{Z_n\}$. Firstly, we study the asymptotic properties of the

harmonic moments $\E(Y_t^{-r})$ of order $r>0$ as $n\rightarrow\infty$. Then, we obtain the large deviations of the Lotka-Negaev estimator of offspring mean.

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