I have a question about the following passage from my book (see screenshot below).
The book says: the $j$th component of $\mathbf{s}Q^n$ is $P(X_n = j)$
However shouldn't it say: the $j$th component of $\mathbf{s}Q^n$ is $P(X_n = j \mid \text{given $X_0$ has pmf $\mathbf{s}$})$
It seems to me that $\mathbf{s}Q^n$ does not yield the unconditional distribution of $X_n$ since it depends on how you set the pmf for $\mathbf{s}$.
To get the unconditional distribution $P(X_n =j)$ then you have to let $n \rightarrow \infty$ and you need your chain to settle on something (ergodic is the term maybe).
Can you help clarify my confusion? Thanks for your patience and help.
Passage from book
