= Independent stationary proposal lemma
{title2=$q(x,y)=\pi(y)\Rightarrow P(x,y)=\pi(y)$}
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.
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