Lower deviation probabilities for supercritical Galton–Watson processes

By K. Fleischmann, Vitali Wachtel. 2005

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Abstract There is a well-known sequence of constants c n describing the growth of supercritical Galton–Watson processes Z n . By lower deviation 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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a21bd1fbe1b5de8c6421ad5dfc5253ec8f6eb061