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Extremal dependence of random scale constructions

  • Sebastian Engelke
  • Thomas Opitz
  • Jennifer WadsworthEmail author
Open Access


A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction (X1,X2) = R(W1,W2), with non-degenerate R > 0 independent of (W1,W2). Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of R and (W1,W2), the shape of the support of (W1,W2), and dependence between (W1,W2). When R is distinctly lighter tailed than (W1,W2), the extremal dependence of (X1,X2) is typically the same as that of (W1,W2), whereas similar or heavier tails for R compared to (W1,W2) typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of (X1,X2) possible in such cases when (W1,W2) exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.


Copula Extreme value theory Residual tail dependence Tail dependence 

AMS 2000 Subject Classifications

60G70 60E05 62H20 



Financial support from the Swiss National Science Foundation grant 200021-166274 (Sebastian Engelke) and UK Engineering and Physical Sciences Research Council grant EP/P002838/1 (Jennifer Wadsworth) is gratefully acknowledged. Thomas Opitz was partially funded by the French national programme LEFE/INSU.


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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Research Center for StatisticsUniversity of GenevaGenevaSwitzerland
  2. 2.Biostatistics and Spatial Processes, INRAAvignonFrance
  3. 3.Department of Mathematics and Statistics, Fylde CollegeLancaster UniversityLancasterUK

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