Copulas provide a powerful and flexible tool for modeling the dependence structure of random vectors, and they have many applications in finance, insurance, engineering, hydrology, and other fields. One well-known class of copulas in two dimensions is the Farlie–Gumbel–Morgenstern (FGM) copula, since its simple analytic shape enables closed-form solutions to many problems in applied probability. However, the classical definition of the high-dimensional FGM copula does not enable a straightforward understanding of the effect of the copula parameters on the dependence, nor a geometric understanding of their admissible range. We circumvent this issue by analyzing the FGM copula from a probabilistic approach based on multivariate Bernoulli distributions. This paper examines high-dimensional exchangeable FGM copulas, a subclass of FGM copulas. We show that the dependence parameters of exchangeable FGM copulas can be expressed as a convex hull of a finite number of extreme points. We also leverage the probabilistic interpretation to develop efficient sampling and estimating procedures and provide a simulation study. Throughout, we discover geometric interpretations of the copula parameters that assist one in decoding the dependence of high-dimensional exchangeable FGM copulas.

We consider a simple random walk on started at the origin and stopped on its first exit time from . Write L in the form with and N an integer going to infinity in such a way that for some real constant . Our main result is that for , the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level , where is the exit time of a Brownian motion from the unit cube that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).