Remarks:

A sample produced by the above algorithm differs from an iid sample. For one thing, such a sample may involve repeated occurrences of the same value, since rejection of $Y_t$ leads to repetition of $X^{(t)}$ at time $t+1$ (an impossible occurrence in absolutely continuous iid settings)

Minimal regularity conditions on both $f$ and the conditional distribution $q$ for $f$ to be the limiting distribution of the chain $X^{(t)}$: $\cup_{x\in \mathrm{supp}\, f}\mathrm{supp}\, q(y\mid x)\supset \mathrm{supp}\,,f$.

$f$ is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.

$K(x,y) = \rho(x,y)q(y\mid x)+(1-r(x))\delta_x(y)\,.$

The MH Markov chain has, by construction, an invariant probability distribution $f$, if it is also an aperiodic Harris chain, then we can apply ergodic theorem to establish a convergence result.

A sufficient condition to be aperiodic: allow events such as $\{X^{(t+1)}=X^{(t)}\}$.

Property of irreducibility: sufficient conditions such as positivity of the conditional density $q$.

If the MH chain is $f$-irreducible, it is Harris recurrent.

A somewhat less restrictive condition for irreducibility and aperiodicity.

In the following sections, let's introduce other versions of Metropolis-Hastings.