The Annals of Statistics

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

Axel Bücher, Johan Segers, and Stanislav Volgushev

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In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain degree of smoothness. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypo-convergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies $L^{p}$ convergence. For the examples mentioned above, weak convergence with respect to the new metric is established in situations where it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties of resampling procedures and goodness-of-fit tests.

Article information

Ann. Statist., Volume 42, Number 4 (2014), 1598-1634.

First available in Project Euclid: 7 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 62G30: Order statistics; empirical distribution functions
Secondary: 62G32: Statistics of extreme values; tail inference 62M09: Non-Markovian processes: estimation

Bootstrap copula epigraph hypograph linear regression local alternative power curve residual empirical process stable tail dependence function weak convergence


Bücher, Axel; Segers, Johan; Volgushev, Stanislav. When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs. Ann. Statist. 42 (2014), no. 4, 1598--1634. doi:10.1214/14-AOS1237.

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Supplemental materials

  • Supplementary material: Supplement to: “When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs”. In the supplement, missing proofs for the results in this paper are given.