Criticality in unbounded-types branching processes
  Conditions for almost sure extinction are studied in discrete time branching processes with an infinite number of types. It is not assumed that the expected number of children is a bounded function of the parent's type. There might also be no integer m such that there is a lower positive bound, uniform over the ancestor's type, for the probability that a population is extinct at the m-th generation. A weaker condition than the existence of such an m is seen to lead to extinction almost surely if the sequence of expected generation sizes does not tend to infinity. Some criteria for a positive probability of nonextinction are given. Examples are provided by extending to our setting two applications, namely Leslie population dynamics and processes arising in continuum percolation in which the offsprings follow Poisson point distributions. 
G.T. Tetzlaff