Large deviation rates for Markov branching processes

Large deviation rates are determined for quantities associated with a Markov branching process (Xt:t ≥ 0) having offspring mean m (1,∞) and split rate a > 0. The principal quantities examined are P(|Xt+τ/Xt -exp(a(m - 1)t)| > ) and P(|W - Wt| > ), where W is the almost sure limit of an appropriately normed version of Xt. Modifications and conditional versions are examined. Some of this requires determination of the asymptotic behavior of harmonic moments E(Xt-r).