= Solution
Here $q(x,y)=p(y)$, with fresh proposal randomness independent of the past. For $x,y\in S$, the <Metropolis–Hastings acceptance probability> is
$$
\alpha(x,y)=\min\left\{1,\frac{p(y)p(x)}{p(x)p(y)}\right\}=1.
$$
Thus $P(x,y)=p(y)$ is independent of $x$, and $X_1,\ldots,X_N$ are <independent and identically distributed random variables> with <probability distribution> $p$, even if $X_0$ was chosen arbitrarily in $S$. \b[The method becomes ordinary <Monte Carlo method> sampling from $p$.] The average of $f(X_1),\ldots,f(X_N)$ is an <unbiased estimator>, with the <variance> stated in part (a). Including an arbitrary initial state would spoil this finite-sample property. This is the <independent stationary proposal lemma>.
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