How to prove a Branching Process is transient for all $k>0$ and calculate extinction probability

Question

Consider a branching process $\{z_n\}$. $z_0=0$, $z_{n+1}=\sum_{j=1}^{z_n}\xi_{z_nj}$ and the extinction probability is $\rho$. $\xi_{ij}$ are i.i.d.

(i). If $\mathbb{P}(\xi_{ij}=1)<1$, prove that all $k>0$ is transient. (Hint: consider two cases: $\mathbb{P}(\xi_{ij}=0)=0$ and $\mathbb{P}(\xi_{ij}=0)\in(0,1]$)

(ii). Based on the results in (i), calculate $\lim_{n\to \infty}\mathbb{P}(z_n \to \infty)$.

For question (i), I can prove that under the first case, i.e., $\mathbb{P}(\xi_{ij}=1)<1$ and $\mathbb{P}(\xi_{ij}=0)=0$. Define $$f_{kk}^{(t)}=\mathbb{P}(z_1\ne k, z_2\ne k, \ldots, z_t= k | z_0=k)$$ and under the first case we have $f_{kk}^{(1)}=\left[\mathbb{P}(\xi_{ij}=1)\right]^k<1$ and $f_{kk}^{(t)}=0, t>1$. Hence under the first case $$\sum_{t=1}^{\infty}f_{kk}^{(t)}<1$$ and all $k>0$ is transient state. But what if the second case where $\mathbb{P}(\xi_{ij}=0)\in(0,1]$? It seems unhelpful to calculate $\sum_{t=1}^{\infty}f_{kk}^{(t)}$

As for question (ii), I guess the answer is $1-\rho$. However, I don't know how to derive this result from question (i). Moreover, I am really curious about the notation $\mathbb{P}(z_n \to \infty)$. Can someone help? Thanks a lot :)

Answer 1

(i) when $b:=P(\xi_{i,j}=0)>0$: Because $$ P(z_{n+1}=0\,|\,z_n=k)=b^k>0 $$ and state $0$ is absorbing, the chain has no states other that $0$ that are recurrent. So each state $k\ge 1$ must be transient.

(ii) from (i): First, $$ \{z_n\to\infty\} = \cap_{M=1}^\infty \cup_{N=1}^\infty\cap_{n= N}^\infty\{z_n> M\}. $$ Because state $k>0$ is transient, $$ P(z_n=k \hbox{ i.o.})=0. $$ That is, $$ P\left( \cap_{N=1}^\infty \cup_{n=N}^\infty\{z_n=k\}\right)=0. $$ Consequently $$ P(z_n\in\{1,2,\ldots,M\}\hbox{ i.o.}) =0, $$ for each $M\ge 1$. Let $B:=\{z_n\to 0\}$ denote the event that the process goes extinct, so $P(B) =\rho$. Then: $$ \eqalign{ P\left(B^c\cap\{z_n\to\infty\}^c\right) &=P\left(\cup_{M=1}^\infty\{z_n\in\{1,2,\ldots,M\}\hbox{ i.o.}\}\right)\cr &\le\sum_{M=1}^\infty P\left(z_n\in\{1,2,\ldots,M\}\hbox{ i.o.}\right)\cr &=0.\cr } $$ It follows that $P(z_n\to\infty) =1-P(z_n\to 0) =1-\rho$.