Show that the branching process $\{X_n\}_{n\ge1}$ is a MC with state space $\mathbb{N}^+\cup\{0\}$. Find its transition matrix.
The first part is easy and I could do it in two steps-
- Proved that $\mathbb{P}[X_{n+1}=s_{n+1}|X_n=s_n]=\mathbb{P}[X_{n+1}=s_{n+1}]$
- Proved that $\mathbb{P}[X_{n+1}=s_{n+1}|X_n=s_n,...X_1=s_1]=\mathbb{P}[X_{n+1}=s_{n+1}]$
using the independence of $\{Y_{ij}\}_{i,j}$ which is the number of offsprings of the $j$th indiviudal in $i$th generation.
For transition matrix, I need to find $\mathbb{P}[X_n=s_n|X_m=s_m]$ for any $n,m$. Using $$\mathbb{P}[X_{n+1}=s_{n+1}|X_n=s_n]=\mathbb{P}\left[\sum_{j=1}^{s_n}Y_{n,j}|X_n=s_n\right]$$ how do I calculate what is needed?