Tail Behaviour of Copulas

The study and modeling of interdependencies between extreme events is crucial for many applications of probability theory and statistics. Thanks to Sklar’s Theorem such tasks decompose into the study of the tail behaviour of the marginal univariate distributions and of the tail (i.e. corner) behaviour of the corresponding copulas. In this chapter we will deal with the second "subproblem". There are several approaches known in the literature.We shall deal with the one based on the tail expansion of copulas near the vertex (0, …, 0) of the unit multicube.We present the notions related to the tail expansion – leading parts, tail dependence functions and limiting invariant measures.We briefly discuss their properties and characterizations and provide several examples of the tail behaviour of copulas. Next we show relations between the tail expansion method and other approaches to the tail behaviour of copulas like the ones based on conditional copulas or associated extreme value copulas. At the end we present possible applications of the notion of tail expansions to quantitative finance, especially to risk measurement.