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We study the distribution of the maximal jump of continuous-state branchingprocesses. Several exact expressions and explicit asymptotics of both the localmaximal jump and the global maximal jump are obtained. We also compare thedistribution of the maximal jump and the L\'{e}vy measure to get severalabsolute continuity results. Then we study local convergence ofcontinuous-state branching processes under various conditionings. We obtaincomplete results under the conditioning of large maximal jump, and partialresults under two other conditionings, which are, the conditioning of largewidth, and, the conditioning of large total mass.

The maximal jump and local convergence of continuous-state branching processes