a) Let the branching process with offspring distribution given by {pn}. We will make process into the irreducible Markov chain by asserting that if population ever dies out, then next generation will have 1 new individual [in other words, p(0,1) = 1]. For which {pn} will chain be positive recurrent, null recurrent, transient?
b) Let the given variation of branching process. At each time n, every individual produces offspring independently using offspring distribution {pn}, and then individual dies with probability q Î (0,1). Therefore, every individual reproduces j times where j is lifetime of individual. For which values of q, {pn} do we have ultimate extinction with probability 1?