Independent stationary proposal lemma ($q(x,y)=\pi (y)\Rightarrow P(x,y)=\pi (y)$)

If a Metropolis–Hastings algorithm proposes fresh draws from its target stationary distribution $\pi$, its acceptance ratio on the positive support is $\pi (y)\pi (x)/(\pi (x)\pi (y))=1$. Therefore every proposal is accepted, and $P(x,y)=\pi (y)$. From the first update onward the recorded states are independent and identically distributed random variables with law $\pi$, regardless of the initial state in the support. An arbitrary initial state is excluded when claiming an unbiased finite-sample average.