I got the following question about coupling yesterday in class: why can't we just use statistical coupling to prove the ergodic theorem (i.e., can't we leave out the second part about grand coupling, leaving the chains X and Y to move independently until the end of time)?
The answer is the following: in this case, it still holds for every n that
|| mu P^n - nu P^n ||_TV <= P (X_n != Y_n)
but the problem is that P (X_n != Y_n) might now be greater than P (tau_couple > n); indeed, since the chains are independent all along, the fact that X_n != Y_n does not ensure anymore that coupling has not yet happened. And more than that, in this case, once we get close to the stationary distribution pi, P (X_n != Y_n) approaches a given value which is not zero.
