Abstract
A branching process Z is said to be non conservative if it hits ∞ in a finite time with positive probability. It is well known that this happens if and only if the branching mechanism φ of Z satisfies ∫0+dλ∕|φ(λ)|<∞. We construct on the same probability space a family of conservative continuous state branching processes Z(ε), ε≥0, each process Z(ε) having φ(ε)(λ)=φ(λ+ε)−φ(ε) as branching mechanism, and such that the family Z(ε), ε≥0 converges a.s. to Z, as ε→0. Then we study the speed of convergence of Z(ε), when ε→0, referred to here as the explosion speed. More specifically, we characterize the functions f with limε→0f(ε)=∞ and such that the first passage times σε=inf{t:Zt(ε)≥f(ε)} converge toward the explosion time of Z. Necessary and sufficient conditions are obtained for the weak convergence and convergence in L1. Then we give a sufficient condition for the almost sure convergence.
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Claims 0
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