Coupling methods in approximations

Let X and Y be two arbitrary k ‐dimensional discrete random vectors, for k ≥ 1. We prove that there exists a coupling method which minimizes P ( X ≠ Y ). This result is used to find the least upper bound for the metric d ( X, Y ) = sup A | P ( X ∈ A ) − P ( Y ∈ A )| and to derive the inequality d(Σ X i , Σ Y i ) ≤ Σ d( X i , Y i ). We thus obtain a unified method to measure the disparity between the distributions of sums of independent random vectors. Several examples are given.