Coupled Samples in Simulation

Assume that we wish to generate two samples of n independent identically distributed random variables, (X₁,…, X_n) and (Y₁,…, Y_n), where X₁ and Y₁ have densities f and g, respectively. If these samples are used in a simulation, and f is close to g, it is sometimes desirable to have close simulation results. This can be achieved by insisting that both samples agree in most of their components, that is, X_i=Y_i for as many i as possible under the given distributional constraints. Samples with this property are said to be optimally coupled. In this paper, we propose and study various methods of coupling two samples, a sequence of samples and an infinite family of samples.