A primer on the characterization of the exchangeable Marshall–Olkin copula via monotone sequences

Abstract

While derivations of the characterization of the $d$-variate exchangeable Marshall–Olkin copula via $d$-monotone sequences relying on basic knowledge in probability theory exist in the literature, they contain a myriad of unnecessary relatively complicated computations. We revisit this issue and provide proofs where all undesired artefacts are removed, thereby exposing the simplicity of the characterization. In particular, we give an insightful analytical derivation of the monotonicity conditions based on the monotonicity properties of the survival probabilities.