Geodesic
A multivariate Bahadur--Kiefer representation for the empirical copula process
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 14–2, Tome 364 (2009), pp. 120-147

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We provide a multivariate extension of the Kiefer (1970) strong limit law for the Bahadur–Kiefer reperesentation. This allows us to derive optimal rates for the strong approximation of empirical copula processes by sequences of Gaussian processes. We also provide a full characterization of empirical copulas in a general framework. Bibl. – 30 titles. 
@article{ZNSL_2009_364_a5,
     author = {P. Deheuvels},
     title = {A multivariate {Bahadur--Kiefer} representation for the empirical copula process},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {120--147},
     publisher = {mathdoc},
     volume = {364},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_364_a5/}
}
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P. Deheuvels. A multivariate Bahadur--Kiefer representation for the empirical copula process. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 14–2, Tome 364 (2009), pp. 120-147. http://geodesic.mathdoc.fr/item/ZNSL_2009_364_a5/