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 leads to repetition of at time (an impossible occurrence in absolutely continuous iid settings)
Minimal regularity conditions on both and the conditional distribution for to be the limiting distribution of the chain : .
is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.
A sufficient condition to be aperiodic: allow events such as .
Property of irreducibility: sufficient conditions such as positivity of the conditional density .
If the MH chain is -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.