Geodesic
Limit theorems for probabilities of large deviations of a Galton-Watson process
Diskretnaya Matematika, Tome 15 (2003) no. 1, pp. 3-27

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We prove local and integral limit theorems for large deviations of Cramer type for a critical Galton–Watson branching process under the assumption that the radius of convergence of the generating function of the progeny is strictly greater than one. The proof is based on a modified Cramer approach which consists of construction of an auxiliary non-homogeneous in time branching process.This research was supported by the Russian Foundation for Basic Research, grant 02–01–01252, and by INTAS, grants 99–01317, 00–265. 
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     title = {Limit theorems for probabilities of large deviations of a {Galton-Watson} process},
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S. V. Nagaev; V. I. Vakhtel'. Limit theorems for probabilities of large deviations of a Galton-Watson process. Diskretnaya Matematika, Tome 15 (2003) no. 1, pp. 3-27. http://geodesic.mathdoc.fr/item/DM_2003_15_1_a0/