I'm interested in a Markov chain with, say, 5 kernels $P_1, \dots, P_5$ that cycle deterministically over the time steps of the chain. This is nominally a time-inhomogeneous chain.
According to this quote from a textbook [0], such a chain can be made time-homogeneous.
Quote:
5.13 Periodicity in time. This is a special case of time-inhomogeneity which is encountered often in applied work where seasonal effects or the day-of-the-week effects and the like affect the transition probabilities. For instance, if $X_n$ is to denote the inventory level for some item at the end of day $n$, then we would expect $P_n(x, A)$ to depend on $n$ only through whether $n$ is a Monday or Tuesday and so on. In such cases, there is an integer $d ≥ 2$ such that the sequence of transition kernels $P_n$ in 5.11 has the form
$$(P_1, P_2, . . .) = (P_1, P_2, . . . , P_d, P_1, P_2, . . . , P_d, . . .).$$
The corresponding time-inhomogeneous chain $X$ can be rendered time-homogeneous by incorporating periodicity into the state space: Let $D = \{1, 2, . . . , d\}, \mathscr D = 2^D, \hat E = D × E, \mathscr{\hat E} = \mathscr D ⊗ \mathscr E$, and define $\hat P$ as a Markov kernel on $(\hat E, \hat{\mathscr E})$ that satisfies
$$\hat P (y, B) = P_j (x, A) \text{ if } y = (i, x), B = \{j\} × A, j = (1 + i) \mod d.$$
Then, $\hat X_n = (n \mod d, X_n), n ∈ \mathbb N$, form a time-homogeneous Markov chain with state space $( \hat E, \hat{\mathscr E})$ and transition kernel $\hat P$. Again, this is not canonical.
Question:
Can someone help me understand this construction? Let's say I have 5 kernels $P_1, \dots, P_5$. If we consider $\hat P (y, B)$, then let's say $y = (3, x) \in \hat E = D \times E$. So if we start from time $n = 1$, then times $3, 8, 13, \dots$ should use the kernel $P_3$.
But why do we use $P_j(x, A)$, where $j = (1 + i) \mod d$? Then wouldn't we have
$$j = (1 + 3) \mod 5 = 4 \mod 5 = 4?$$
I don't understand this construction at all. Shouldn't the modulus operator should be applied to the time step $n$, not $i \in D = \{1, 2, \dots, d\}$?
I appreciate any help.
0: Probability and Stochastics, Çınlar.