The Annals of Statistics

Limit theorems for empirical processes of cluster functionals

Holger Drees and Holger Rootzén

Full-text: Open access


Let (Xn,i)1≤in,n∈ℕ be a triangular array of row-wise stationary ℝd-valued random variables. We use a “blocks method” to define clusters of extreme values: the rows of (Xn,i) are divided into mn blocks (Yn,j), and if a block contains at least one extreme value, the block is considered to contain a cluster. The cluster starts at the first extreme value in the block and ends at the last one. The main results are uniform central limit theorems for empirical processes $Z_{n}(f):=\frac{1}{\sqrt{nv_{n}}}\sum_{j=1}^{m_{n}}(f(Y_{n,j})-Ef(Y_{n,j}))$, for vn = P{Xn,i ≠ 0} and f belonging to classes of cluster functionals, that is, functions of the blocks Yn,j which only depend on the cluster values and which are equal to 0 if Yn,j does not contain a cluster. Conditions for finite-dimensional convergence include β-mixing, suitable Lindeberg conditions and convergence of covariances. To obtain full uniform convergence, we use either “bracketing entropy” or bounds on covering numbers with respect to a random semi-metric. The latter makes it possible to bring the powerful Vapnik–Červonenkis theory to bear. Applications include multivariate tail empirical processes and empirical processes of cluster values and of order statistics in clusters. Although our main field of applications is the analysis of extreme values, the theory can be applied more generally to rare events occurring, for example, in nonparametric curve estimation.

Article information

Ann. Statist., Volume 38, Number 4 (2010), 2145-2186.

First available in Project Euclid: 11 July 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F17: Functional limit theorems; invariance principles 62G32: Statistics of extreme values; tail inference

Absolute regularity block bootstrap clustering of extremes extremes local empirical processes rare events tail distribution function uniform central limit theorem


Drees, Holger; Rootzén, Holger. Limit theorems for empirical processes of cluster functionals. Ann. Statist. 38 (2010), no. 4, 2145--2186. doi:10.1214/09-AOS788.

Export citation


  • Bortot, P. and Tawn, J. A. (1998). Models for the extremes of Markov chains. Biometrika 85 851–867.
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory. Springer, New York.
  • Drees, H. (2000). Weighted approximations of tail processes for β-mixing random variables. Ann. Appl. Probab. 10 1274–1301.
  • Drees, H. (2002). Tail empirical processes under mixing conditions. In Empirical Process Techniques for Dependent Data (H. G. Dehling, T. Mikosch and M. Sørensen, eds.) 325–342. Birkhäuser, Boston.
  • Drees, H. (2003). Extreme quantile estimation for dependent data with applications to finance. Bernoulli 9 617–657.
  • Drees, H. (2010). Bias correction for blocks estimators of the extremal index. Univ. Hamburg. Preprint.
  • Dudley, R. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press, Cambridge.
  • Eberlein, E. (1984). Weak convergence of partial sums of absolutely regular sequences. Statist. Probab. Lett. 2 291–293.
  • Einmahl, J. (1997). Poisson and Gaussian approximation of weighted local empirical processes. Stochastic Process. Appl. 70 31–58.
  • Fabian, V. (1970). On uniform convergence of measures. Z. Wahrsch. Verw. Gebiete 15 139–143.
  • Giné, E. and Mason, D. M. (2008). Uniform in bandwidth estimation of integral functionals of the density function. Scand. J. Statist. 35 739–761.
  • Giné, E., Mason, D. M. and Zaitsev, A. Y. (2003). The L1-norm density estimator process. Ann. Probab. 31 719–768.
  • Hahn, M. G. (1977). Conditions for sample-continuity and the central limit theorem. Ann. Probab. 5 351–360.
  • Leadbetter, M. R. (1995). On high level exceedance modeling and tail inference. J. Statist. Plann. Inference 45 247–260.
  • Leadbetter, M. R. and Rootzén, H. (1993). On central limit theory for families of strongly mixing additive random functions. In Stochastic Processes: A Festschrift in Honour of Gopinath Kallianpur (S. Cambanis et al., eds.) 211–223. Springer, New York.
  • Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin.
  • Rootzén, H. (1995). The tail empirical process for stationary sequences. Preprint, Chalmers Univ.
  • Rootzén, H. (2009). Weak convergence of the tail empirical function for dependent sequences. Stochastic Process. Appl. 119 468–490.
  • Rootzén, H., Leadbetter, M. R. and de Haan, L. (1990). Tail and quantile estimators for strongly mixing stationary processes. Report, Dept. Statistics, Univ. North Carolina.
  • Rootzén, H., Leadbetter, M. R. and de Haan, L. (1998). On the distribution of tail array sums for strongly mixing stationary sequences. Ann. Appl. Probab. 8 868–885.
  • Segers, J. (2003). Functionals of clusters of extremes. Adv. in Appl. Probab. 35 1028–1045.
  • Sisson, S. and Coles, S. (2003). Modelling dependence uncertainty in the extremes of Markov chains. Extremes 6 283–300.
  • Stott, P. A., Stone, D. A. and Allen, M. R. (2004). Human contribution to the European heatwave of 2003. Nature 432 610–613.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • Wellner, J. A. and Zhang, Y. (2000). Two estimators of the mean of a counting process with panel count data. Ann. Statist. 28 779–814.
  • Wellner, J. A. and Zhang, Y. (2007). Two likelihood-bsed semiparametric estimation methods for panel count data with covariates Ann. Statist. 35 2106–2142.
  • Yun, S. (2000). The distribution of cluster functionals of extreme events in a dth-order Markov chain. J. Appl. Probab. 37 29–44.