# Copula Theory: an Introduction

Topics: Probability theory, Random variable, Probability Pages: 17 (5309 words) Published: May 11, 2013
Copula Theory: an Introduction
Fabrizio Durante and Carlo Sempi

Abstract In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various ﬁelds, and several methods of constructing multivariate copulas.

This version: September 14, 2009

1 Historical introduction
The history of copulas may be said to begin with Fr´ chet [69]. He studied the fole lowing problem, which is stated here in dimension 2: given the distribution functions F1 and F2 of two random variables X1 and X2 deﬁned on the same probability space Ω ,F,P , what can be said about the set Γ F1 ,F2 of the bivariate d.f.’s whose marginals are F1 and F2 ? It is immediate to note that the set Γ F1 ,F2 , now called the Fr´ chet class of F1 and F2 , is not empty since, if X1 and X2 are independent, e then the distribution function x1 ,x2 F x1 ,x2 = F1 x1 F2 x2 always belongs to Γ F1 ,F2 . But, it was not clear which the other elements of Γ F1 ,F2 were. Preliminary studies about this problem were conducted in [64, 70, 88] (see also [30, 181] for a historical overview). But, in 1959, Sklar obtained the deepest result in this respect, by introducing the notion, and the name, of a copula, and proving the theorem that now bears his name [191]. In his own words [193]:

Fabrizio Durante Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Austria e-mail: fabrizio.durante@jku.at Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi” Universit` del Salento, Lecce, Italy e-mail: carlo.sempi@unisalento.it a

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The proof of Sklar’s theorem was not given in [191], but a sketch of it was provided in [192] (see also [184]), so that for a few years practitioners in the ﬁeld had to reconstruct it relying on the hand–written notes by Sklar himself; this was the case, for instance, of the second author. It should be also mentioned that some “indirect” proofs of Sklar’s theorem (without mentioning copula) were later discovered by Moore and Spruill [144] and Deheuvels [36] For about 15 years, all the...