Assumption: Let $\{X_n^{(1)}\}$ and $\{X_n^{(2)}\}$ be two Markov chains with the same transition matrix $P$. Suppose the two chains evolve independently at each step, and a single chain has stationary measure $\pi$.

Theorem: The joint chain

$$ \{(X_n^{(1)}, X_n^{(2)})\} $$

on the state space $S \times S$ has stationary measure given by the product measure $\pi \otimes \pi$. Its components are defined by

$$ (\pi \otimes \pi)_{(i,j)} = \pi_i \pi_j, \quad \forall (i,j) \in S \times S. $$

Verification: $$ \sum_{(k,l) \in S \times S} (\pi_k \pi_l) p_{ki} p_{lj}

\left(\sum_k \pi_k p_{ki}\right)\left(\sum_l \pi_l p_{lj}\right)

\pi_i \pi_j. $$