Lower deviation probabilities for supercritical Galton–Watson processes

There is a well-known sequence of constants cn describing the growth of supercritical Galton–Watson processes Zn. By 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.

RésuméLes auteurs présentent une analyse détaillée des probabilités de déviations inférieures. Ces dernières sont nécessaires à la description du rapport Zn+1/Zn.