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We extend recent results of Schuh on the convergence of 
, where α ≧ 0 and Sn is the sum of n i.i.d. positive random variables to sums of the form 
for a large class of functions g, give simpler proofs than those of Schuh, and derive reformulations of the explosion criteria for Markov branching processes with discrete and continuous state space.
We give necessary and sufficient conditions for 
in terms of 
, where Sn is the sum of n i.i.d. random variables with values in]0, ∞[, and A ≧ 0. We use these results to give a probabilistic proof of the ‘explosion criterion' for continuous-time Markov branching processes, which is usually shown analytically.
