Small positive values and lower large deviations for supercritical branching processes in random environment

Abstract. We give explicit formulae for most likely paths to extinction in simple branching models when initial population is large. In discrete time we study the Galton-Watson process and in continuous time the Branching diffusion. The most likely paths are found with the help of the Large Deviation Principle (LDP). We also find asymptotics for the extinction probability, which gives a new expression in continuous time and recovers the known formula in discrete time. Due to the non-negativity of the processes, the proof of LDP at the point of extinction uses a nonstandard argument of independent interest. 1. Introduction and

Abstract. There is a well-known sequence of constants cn describing the growth of supercritical Galton-Watson processes Zn. With “lower deviation probabilities ” we refer to P(Zn = kn) with kn = o(cn) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Zn+1/Zn. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramér method for proving large deviations of sums of independent variables to our needs. Contents

Abstract. In this paper we study the large deviation behavior of sums of i.i.d. random variables Xi defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX 2 1 and EZ1 log Z1. The underlying interplay of the partial sums of the Xi and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results