Shift-coupling
Let \(X,X',Y,Y'\) be discrete time stochastic processes. The pair \((Y,Y')\) is a coupling of \(X\) and \(X'\) if \(Y\) has the same distribution as \(X\) and \(Y'\) has the same distribution as \(X'\). A pair \((S,T)\) of random times is a shift coupling if \(Y_{S+n}=Y_{T+n}'\) for \(n=0,1,2,\dots\). This extends the usual couplings where \(S=T\), a further extension is given to coupling with respect to \(\sigma\)-fields. The authors show that shift-coupling is related to invariant \(\sigma\)- fields similar to the way that coupling is related to tail \(\sigma\)- fields. Also, couplings are considered that are maximal in the sense that the tails of the coupling time distribution provide sharp upper bounds on the distance of the \(X\)- and \(X'\)-distributions.
scientific article; zbMATH DE number 167124