Lemma ($W_{n}\sim \mbox{Markov}(\tilde\theta,\tilde P)$)

$\{W_{i}\}_{i=0}^\infty$ is a MC with the state space $S^{2}$, initial distribution $\tilde\theta$ and transition matrix $\tilde P$: $$\begin{align*} &\tilde\theta_{(i,j)}=\lambda_{i}\mu_{j}&\mbox{for any }i,j\in S\\ &\tilde P_{(i,j),(i',j')}=P_{i,i'}P_{j,j'}&\mbox{for any }i,i'j,j'\in S \end{align*}$$

Lemma (Irreducibility of the JMC)

If $P$ is irreducible and aperiodic, then $\tilde P$ is irreducible

Lemma (Invariant Distribution of the JMC)

Let $\pi$ be an Invariant Distribution for $P$. Define $$\tilde\pi_{(i,j)}=\pi_{i}\pi_{j} \mbox{ for }i,j\in S$$Then $\tilde\pi$ is invariant for $\tilde P$.

Lemma (Coupling lemma)

Let $T=\inf\{n\ge0:X_{n}=Y_{n}\}$. Define for each $n\ge0$ \begin{align*} {X_{n}'=\left\{\begin{array}\ X_{n} & \mbox{if }n<T\\ Y_n&\mbox{if }n\ge T\end{array}\right.} \ \ \ \ \ {Y_{n}'=\left\{\begin{array}\ Y_{n} & \mbox{if }n<T\\ X_n&\mbox{if }n\ge T\end{array}\right.} \end{align*} Let $W_{n}'=(X_{n}',Y_{n}')$ for $n\ge0$

1. $\{W_{n}':n\ge 0\}$ have the same distribution as JMC $\{W_{n}:n\ge 0\}$
2. Consequently, for any integer $n$, $$\sup_{j\in S}|P(X_{n}=j)-P(Y_{n}=j)|\le P(T>n)$$