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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.

Limit theorems for probabilities of large deviations of a Galton–Watson process