Let , be the size of a population of bacteria at time step n. At each step each bacteria produces a number of offspring and dies. The number of offspring is independent for each bacteria and is distributed according to the Poisson law with parameter .
Assuming that , the problem is to find the probability that the population will eventually die, e.g. for some .
Note there is a hint: find such that is a martingale.
This is a Markovian process. Finding c can be done by evaluating the martingale property at . Through one finds . The martingale enables to use the Optional Sampling Theorem with a stopping time.
But which one? The case , with if the event never occurs raises the question of what is . There is no condition that would allow some form of convergence of the sequence. Is there any way to solve using such stopping time?
Another approach would be to set another barrier, in the form of the stopping time , and then take the limit . With this one gets a value for the probability for some , which I suspect is the right answer.
But this would assume that the process is sure to reach any given bound set, if it doesn't go to zero. This has surely to do with the nature of the process. In a usual "birth-death" process, one would say that the renewal rate being , we know . Is there any way to make a statement like this here?
Thanks for your help.