Probabilistic Inequalities for the Galton--Watson Processes

Probabilistic inequalities are deduced for all values of a breeding coefficient $A$ of a branching process. Previously, only the case A=1 was considered. Separately the cases are studied such that the Cramer condition is fulfilled and there exists only the finite number of moments. Thereafter one of the obtained inequalities is applied for deducing the Rosenthal-type moment inequality for $A\ge 1$.

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