I'm brand new to Markov chains, and a basic question I'm working on goes:
Consider a branching process $\{X_n\}^\infty_{n=0}$ where $X_0=1, X_n=\sum^{X_{n-1}}_{i=1}Y_{i,n}$ for $n \ge1$, and $\{Y_{i,j}\}_{i\ge1,j\ge1}$ are identically and independently distributed variables of the Binomial distribution $Bin(2,\frac{1}{2})$. What is the one-step transition probability of the Markov chain $\{X_n\}^\infty_{n=0}$?
I understand the idea behind branching process is to model how the species may multiply generation by generation based on the probability distribution of how many individuals that each individual of a generation may produce.
By this understanding, $X_0=1$ would mean that there is 1 individual in the first generation, which then produces up to 2 offspring to the next generation by the distribution $Bin(2,0.5)$, and each then reproduces by the same probability. Is this understanding correct?
Also, in this case what exactly is required by the "one-step transition probabilities"? Is is to find the probability of $P(X_1=0\mid X_0 = 1), P(X_1=1\mid X_0 = 1), P(X_1=2\mid X_0 = 1)$? In this case, that would just be the P.M.F. of $P(X_1=0), P(X_1=1), P(X_1=2)$, is this correct?
The next part of the question then asks for the extinction probability at most by step 2. Does this simply mean $P(X_1=0\mid X_0 = 1)+P(X_2=0\mid X_1 = 1) + P(X_2=0\mid X_1 = 2)$? Is this understanding correct as well? If I were to find the extinction probability for many more steps, say 100, would I therefore require a transition matrix?
Thank you in advance to any kind soul willing to assist, your help would be greatly appreciated!