From Wikipedia
The most common formulation of a branching process is that of the Galton–Watson process. Let $Z_n$ denote the state in period $n$ (often interpreted as the size of generation $n$), and let $X_{n,i}$ be a random variable denoting the number of direct successors of member $i$ in period $n$, where $X_{n,i}$ are independent and identically distributed random variables over all $n ∈ \{0, 1, 2, ...\}$ and $i ∈ \{1, ..., Z_n\}$. Then the recurrence equation is $$Z_{n+1} = \sum_{i=1}^{Z_n} X_{n,i}$$ with $Z_0 = 1$.
Alternatively, one can formulate a branching process as a random walk. Let $S_i$ denote the state in period $i$, and let $X_i$ be a random variable that is iid over all $i$. Then the recurrence equation is $$S_{i+1} = S_i+X_{i+1}-1 = \sum_{j=1}^{i+1} X_j-i$$ with $S_0 = 1$.
I don't see how the two definitions are equivalent, or are they? Are the two processes $(Z_n)$ and $(S_n)$ defined in the two definitions the same? How are the two processes $(X_{n,i})$ and $(X_j)$ in the two definitions related to each other?
Thanks!
