Coupling methods in approximations

Abstract

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(XY). This result is used to find the least upper bound for the metric d(X, Y) = supA|P(XA) − P(YA)| and to derive the inequality d(Σ i =1 n Xi, Σ i =1 n Yi) ≤ Σ i =1 n d(Xi, Yi). We thus obtain a unified method to measure the disparity between the distributions of sums of independent random vectors. Several examples are given.

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