• Bernoulli
  • Volume 18, Number 3 (2012), 764-782.

Asymptotics of empirical copula processes under non-restrictive smoothness assumptions

Johan Segers

Full-text: Open access


Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in Stute’s representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.

Article information

Bernoulli, Volume 18, Number 3 (2012), 764-782.

First available in Project Euclid: 28 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Archimedean copula Brownian bridge empirical copula empirical process extreme-value copula Gaussian copula multiplier central limit theorem tail dependence weak convergence


Segers, Johan. Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18 (2012), no. 3, 764--782. doi:10.3150/11-BEJ387.

Export citation


  • [1] Bücher, A. (2011). Statistical inference for copulas and extremes. Ph.D. thesis, Ruhr-Univ. Bochum.
  • [2] Bücher, A. and Dette, H. (2010). A note on bootstrap approximations for the empirical copula process. Statist. Probab. Lett. 80 1925–1932.
  • [3] Chen, S.X. and Huang, T.M. (2007). Nonparametric estimation of copula functions for dependence modelling. Canad. J. Statist. 35 265–282.
  • [4] Csáki, E. (1975). Some notes on the law of the iterated logarithm for empirical distribution function. In Limit Theorems of Probability Theory (Colloq., Keszthely, 1974). Colloquia Mathematica Societatis János Bolyai 11 47–58. Amsterdam: North-Holland.
  • [5] Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d’indépendance. Acad. Roy. Belg. Bull. Cl. Sci. (5) 65 274–292.
  • [6] Denuit, M. and Scaillet, O. (2004). Nonparametric tests for positive quadrant dependence. Journal of Financial Econometrics 2 422–450.
  • [7] Einmahl, J.H.J. (1987). Multivariate Empirical Processes. CWI Tract 32. Amsterdam: Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica.
  • [8] Einmahl, J.H.J. and Mason, D.M. (1988). Laws of the iterated logarithm in the tails for weighted uniform empirical processes. Ann. Probab. 16 126–141.
  • [9] Fermanian, J.D., Radulović, D. and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli 10 847–860.
  • [10] Gaenssler, P. and Stute, W. (1987). Seminar on Empirical Processes. DMV Seminar 9. Basel: Birkhäuser.
  • [11] Genest, C., Quessy, J.F. and Remillard, B. (2007). Asymptotic local efficiency of Cramér-von Mises tests for multivariate independence. Ann. Statist. 35 166–191.
  • [12] Genest, C. and Rémillard, B. (2004). Tests of independence and randomness based on the empirical copula process. Test 13 335–370.
  • [13] Genest, C. and Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. Ann. Statist. 37 2990–3022.
  • [14] Genest, C. and Segers, J. (2010). On the covariance of the asymptotic empirical copula process. J. Multivariate Anal. 101 1837–1845.
  • [15] Ghoudi, K. and Rémillard, B. (2004). Empirical processes based on pseudo-observations. II. The multivariate case. In Asymptotic Methods in Stochastics. Fields Inst. Commun. 44 381–406. Providence, RI: Amer. Math. Soc.
  • [16] Kiefer, J. (1970). Deviations between the sample quantile process and the sample df. In Nonparametric Techniques in Statistical Inference (Proc. Sympos., Indiana Univ., Bloomington, Ind., 1969) (M. Puri, ed.) 299–319. London: Cambridge Univ. Press.
  • [17] Kojadinovic, I. and Holmes, M. (2009). Tests of independence among continuous random vectors based on Cramér–von Mises functionals of the empirical copula process. J. Multivariate Anal. 100 1137–1154.
  • [18] Kojadinovic, I. and Yan, J. (2010). Nonparametric rank-based tests of bivariate extreme-value dependence. J. Multivariate Anal. 101 2234–2249.
  • [19] Leadbetter, M.R. and Rootzén, H. (1988). Extremal theory for stochastic processes. Ann. Probab. 16 431–478.
  • [20] Mason, D.M. (1981). Bounds for weighted empirical distribution functions. Ann. Probab. 9 881–884.
  • [21] McNeil, A. and Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and 1-norm symmetric distributions. Ann. Statist. 37 3059–3097.
  • [22] Omelka, M., Gijbels, I. and Veraverbeke, N. (2009). Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing. Ann. Statist. 37 3023–3058.
  • [23] Pickands, J. III (1989). Multivariate negative exponential and extreme value distributions. In Extreme Value Theory (Oberwolfach, 1987). Lecture Notes in Statist. 51 262–274. New York: Springer.
  • [24] Rémillard, B. and Scaillet, O. (2009). Testing for equality between two copulas. J. Multivariate Anal. 100 377–386.
  • [25] Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Ann. Statist. 4 912–923.
  • [26] Scaillet, O. (2005). A Kolmogorov–Smirnov type test for positive quadrant dependence. Canad. J. Statist. 33 415–427.
  • [27] Schmid, F., Schmidt, R., Blumentritt, T., Gaisser, S. and Ruppert, M. (2010). Copula-based measures of multivariate association. In Copula Theory and Its Applications (Warsaw, 2009). Lecture Notes in Statistics – Proceedings (P. Jaworski, F. Durante, W. Härdle and T. Rychlik, eds.). Springer, Berlin.
  • [28] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
  • [29] Stute, W. (1984). The oscillation behavior of empirical processes: The multivariate case. Ann. Probab. 12 361–379.
  • [30] Tsukahara, H. (2000). Empirical copulas and some applications. Technical Report 27, The Institute for Economic Studies, Seijo Univ. Available at
  • [31] Tsukahara, H. (2005). Semiparametric estimation in copula models. Canad. J. Statist. 33 357–375.
  • [32] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. New York: Springer.
  • [33] van der Vaart, A.W. and Wellner, J.A. (2007). Empirical processes indexed by estimated functions. In Asymptotics: Particles, Processes and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 55 234–252. Beachwood, OH: IMS.