Extinction probability not starting at 1

Question

Consider a branching process where $X_n$ represents the number of individuals in generation $n$. Suppose that you know the extinction probability when $X_0 =1$:

$$\alpha=P(extinction|X_0=1)$$

Could you please help me prove that $$P(extinction|X_0=k) =\alpha^k$$

Answer 1

The branching proces under condition $X_0=k$ can be interpreted as $k$ independent branching processes each under condition $X_0=1$.

So there will be extinction under condition $X_0=k$ if there is extinction in these $k$ independent processes under condition $X_0=1$.

Then that event has probability: $$P(\text{extinction}\mid X_0=k)=P(\text{extinction}\mid X_0=1)^k$$