Thanks for the advertisement. In the next version, we will make sure to be clearer on the coupling of random walk MH. It is actually very simple!

Consider two MH chains at states X_t and Y_t (forget the time shift of the paper for now), each with Normal random walk proposal. So, normally you would draw X* from N(X_t, Sigma), and Y* from N(Y_t, Sigma), with Sigma being the proposal variance, and then decide to accept or not these as the next states of the chains.

What we propose to do, is to sample (X*,Y*) from the maximal coupling of the two distributions N(X_t, Sigma), and N(Y_t, Sigma). The procedure of Algorithm 4 does that exactly, and this was presumably known forever in the coupling literature (it is definitely in Thorisson’s book).

Then there is a chance that X* would be exactly equal to Y*. In that case, it is possible that both proposals would be accepted as X_t+1 and Y_t+1. If not, there’s always the next step.

There is no boundedness condition for this algorithm; we discuss bounded conditions to theoretically justify that the chains will meet geometrically fast. Intuitively, we only use the fact that some event that can happen with >0 probability at every step will happen geometrically fast.

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