An $n$-step transition kernel is defined as (see e.g. Monte Carlo Statistical Methods by Robert & Casella): $$ K^n(x, A) = \int_\mathcal{X} K^{n-1}(y, A) K(x, dy) $$ I am trying to get an intuitive understanding of this expression.
Intuitively, I would have thought that the $n$-step transition kernel should look something like this (I know this doesn't make any sense)
$$ K^n(x, A) = \int_{A} K^{n-1}(x, B) K(B, dy) $$
My reasoning behind this is:
- We use the first $n-1$ steps to move from the initial point $x$ to some point in the set $B$ (any set)
- Then we use the final kernel to move from a point in $B$ to a point $y\in A$.
Something like this (but correct) would make sense to me because it resembles $$ K(x, A) = \int_A K(x, dy) $$ much more since it is an integration over $A$ rather than $\mathcal{X}$.