Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series

By Victor H. de la Peña, Rustam Ibragimov, Sh. Sharakhmetov. 2006

60 citations recorded by the source corpus.

In this paper, we obtain general representations for the joint dis- tributions and copulas of arbitrary dependent random variables absolutely continuous with respect to the product of given one-dimensional marginal dis- tributions. The characterizations obtained in the paper represent joint distrib- utions of dependent random variables and their copulas as sums of U -statistics in independent random variables. We show that similar results also hold for expectations of arbitrary statistics in dependent random variables. As a corol- lary of the results, we obtain new representations for multivariate divergence measures as well as complete characterizations of important classes of depen- dent random variables that give, in particular, methods for constructing new copulas and modeling different dependence structures. The results obtained in the paper provide a device for reducing the analysis of convergence in distribution of a sum of a double array of dependent random variables to the study of weak convergence for a double array of their inde- pendent copies. Weak convergence in the dependent case is implied by similar asymptotic results under independence together with convergence to zero of one of a series of dependence measures including the multivariate extension of Pearson's correlation, the relative entropy or other multivariate divergence measures. A closely related result involves conditions for convergence in dis- tribution of m-dimensional statistics h(Xt ,X t+1 ,...,X t+m−1) of time series {Xt} in terms of weak convergence of h(ξt ,ξ t+1 ,...,ξ t+m−1), where {ξt} is a sequence of independent copies of Xs, and convergence to zero of measures of intertemporal dependence in {Xt}. The tools used include new sharp estimates for the distance between the distribution function of an arbitrary statistic in dependent random variables and the distribution function of the statistic in independent copies of the random variables in terms of the measures of depen- dence of the random variables. Furthermore, we obtain new sharp complete decoupling moment and probability inequalities for dependent random vari- ables in terms of their dependence characteristics.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/fb28cc6ac3e24cdf768eb6a8331f3841e8973870

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Statistics and Probability

Citations in the corpus

COPULA-BASED CHARACTERIZATIONS FOR HIGHER ORDER MARKOV PROCESSES (2009, methodology): “…of Markov processes of an arbitrary order in Section 2 it follows, in particular, that any copulas and, thus, any U -statisticsbased representations for copula functions in de la Peña et al. (2006) (see Appendix A in this paper) can be used to construct higher order Markov processes.”
International diversification: A copula approach (2011, background): “10See de la Peña, Ibragimov, and Sharakhmetov (2006), Definition 3.1.”