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the multivariate distribution function with standard uniform margins. However not this definition made copula so attractive, but rather the Sklar’s theorem, where the very importance of copulae in the area of multivariate distributions has been restated in an elegant way. AMS 2010 Subject Classification: Primary 62-00; Secondary 62H05 Keywords and Phrases: Copula, multivariate distribution 282 Okhrin Theorem 1.2 (Sklar (1959)) Let F be a multivariate distribution function with margins F1; : : : ;Fd , then there exists the copula C./ such that F.x1; : : : ;xd / D C

function H is best characterized by a unique function -a bivariate copula C, defined everywhere on the unit square 12 by the following relation H(x,y) = C(F(x),G(y)), x)2/GR (see Sklar's Theorem, Sklar (1959)). A bivariate copula is a bivariate cu- mulative distribution function with uniform margins on the unit interval I. 2000 Mathematics Subject Classification: 62E10, 62H05. Key words and phrases: Copulas, positive quadrant dependence, regression, charac- terizing theorems, Lancaster probabilities, diagonal expansion of bivariate density, or- thonormal polynomials

(identifiablity), others with characterization of classes of distributions μγ for a given class of posterior means, often assumed to be linear functions (identification). See for instance: Krishnaji (1974), Korwar (1975, 1977), Xekalaki (1983), Cacoullos and Papageorgiou (1983, 1984), Papageorgiou (1984, 1985), Kyriakoussis and Papageorgiou (1991), Johnson and Kotz (1992), Arnold et al. (1993), Sapatinas (1995), Wesolowski AMS (1991) subject classification: Primary 62H05; Secondary 62E10, 62F15. Key words and phrases: mixtures, posterior mean, identifiability

conditional volatili- ties are modelled using, e.g., GARCH-type processes. For a recent review of multivariate GARCH processes, including Dynamic Conditional Correlation (DCC), Constant Condi- tional Correlation (CCC), Baba, Engle, Kraft, and Kroner (1990) (BEKK), and others, we refer to Silvennoinen and Teräsvirta (2009). These models still assume that the parameters for the process are constant over an entire estimation period. Such an approach has been Corresponding author: Ostap Okhrin AMS 2010 subject classification: Primary 62H12, Secondary 62H05 Keywords and

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