Branching Process; understanding theorem proof

Question

Now I'm reading Durrett's probability book and having trouble understanding the proof of one of the theorems related to Branching Process. (Theorem 5.3.8.)

Let $\xi_i^m,$ $\, i,n\geq 1$ be i.i.d. nonnegative integer-valued random variables and let $Z_n$ be a Galton-Watson process w.r.t. $\xi_i^m$.

Suppose $\mu = \mathbb{E}\xi_i^m = 1$ and $\mathbb{P}(\xi_i^m = 1) < 1$.

Then $\mathbb{P}(Z_n = k ~~ \text{for all} ~~ n \geq N) = 0$ for $k > 0$ and for any positive integer $N$.

The last sentence is what I want to understand.

Any help will be appreicated!

Answer 1

Sketch: Notice that $\mathbb{P} ( Z_{n+1} =k |Z_n=k ) \leq 1-\mathbb{P}(\xi^m_i=0)^k<1$.

From the Markov's Propetry we see \begin{align} \mathbb{P}(Z_n = k ~~ \text{for all} ~~ M \geq n \geq N) &= \mathbb{P}(Z_N=k) \mathbb{P} ( Z_{N+1} =k | Z_N=k )\cdots\mathbb{P}(Z_M=k|Z_{M-1} =k)\\ &\leq\mathbb{P}(Z_N=k) ( 1-\mathbb{P}(\xi^m_i=0)^k)^{(M-N-2)} \end{align} Now taking $M\rightarrow \infty$ yields the result.