A condition for the extinction of a branching process with an absorbing lower barrier (Q1234967)

A branching process with an absorbing lower barrier is considered. This is a Galton-Watson process with the condition that at any generation the number of individuals is greater than a lower barrier or it is equal to zero (i.e. all individuals in populations which are too small die and have no offspring). A necessary and sufficient condition is given for the process to survive with positive probability. (More recently, the author has found a necessary and sufficient condition for the more general problem of ensuring that a Galton-Watson process stay always between two boundaries with positive probability. The result will be published elsewhere.)