Deviations for jumping times of a branching process indexed by a Poisson process (Q2298619)

Summary: 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 Poisson process which is independent of \(\{Z_n \}\). Let \(\tau_n\) be the \(n \)-th jumping time of \(\{Y_t \}\), we obtain that the typical rate of growth for \(\{\tau_n \}\) is \(n / \lambda \), where \(\lambda\) is the intensity of \(\{N_t \}\). Probabilities of deviations \(\left\{\left|n^{- 1} \tau_n - \lambda^{- 1}\right| > \delta\right\}\) are estimated for three types of positive \(\delta \).