TAIL MUTUAL INFORMATION OF VINE COPULAS
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Multivariate mutual information describes the amount of uncertainty among several random variables, whose scale-invariant dependence is captured by the copula of the joint distribution. In this dissertation, we first show that multivariate mutual information and conditional mutual information of a distribution are the same as these of its underlying copula. The invariance properties allows us to separate one-dimensional marginal distributions and multivariate dependence in the context of analyzing mutual information. Second, we derive the recursive formulas of multivariate mutual information for vine copulas in terms of conditional mutual informations of bivariate liking copulas, based on the tree structures. The important part of this dissertation consists of introducing for the first time in the literature and analyzing the tail mutual information of multivariate extremes. Specifically, we introduce upper and lower tail mutual information of a multivariate distribution and prove that the rates at which upper or lower conditional mutual information converge to zero can be expressed in terms of conditional mutual information of tail densities. We also establish the scaling properties of upper or lower tail mutual information and show that the scaling tail indexes are the same as that of the underlying multivariate regularly varying distribution or the underlying copula. Our results, such as explicit expressions on converging rates and scaling tail indexes, lay the foundations for multivariate extreme value analysis, based on mutual information.