In a branching process having X0 = 1 and > 1, prove that 0 is the

In a branching process having X0 = 1 and μ > 1, prove that π0 is the smallest positive number satisfying Equation (4.20).

Hint: Let π be any solution of π = ∞

j=0 π j Pj . Show by mathematical induction that π P{Xn = 0} for all n, and let n → ∞. In using the induction argue that P{Xn = 0} = ∞

j=0

(P{Xn−1 = 0})j Pj