Branching processes extinction (homework)

Question

This is my stochastic process course homework. I can solve (a)(b), which are easy to prove. But I have no idea about (c). Could you give me some idea?

Answer 1

Hints:

• $P(\text{extinction}\mid X_n\ne0)=P(\text{extinction}; X_n\ne0)/P(X_n\ne0)$
• $P(X_n\ne0)\geqslant P(\text{survival})=1-P(\text{extinction})$
• $P(\text{extinction}; X_n\ne0)=P(\text{extinction})-P(X_n=0)$

This yields (if I am not mistaken and to help you check your solution to come) the upper bound $$ P(\text{extinction}\mid X_n\ne0)\leqslant\frac{a}{1-a}\rho^n$$