Lower deviation probabilities for supercritical Galton--Watson processes
- Fleischmann, Klaus
- Wachtel, Vitali
2010 Mathematics Subject Classification
- 60J80 60F10
- Supercritical Galton-Watson process, local limit theorem, large deviation, Cramer transform, concentration function, Schroeder equation, Boettcher equation
There is a well-known sequence of constants c_n describing the growth of supercritical Galton-Watson processes Z_n. With "lower derivation probabilities" we refer to P(Z_n = k_n) with k_n = o(c_n) 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 Z_n+1/Z_n. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramer method for proving large deviations of sums of independent variables to our needs.
- Ann. Inst. H. Poincare Probab. Statist., 43 (2007) pp. 233-255.