A note on the Galambos copula and its associated Bernstein function

Jan-Frederik Mai 1 , 2
  • 1 Lehrstuhl für Finanzmathematik (M13), Technische Universität München, Parkring 11, 85748 Garching-Hochbrück, Germany
  • 2 XAIA Investment GmbH, Sonnenstraße 19, 80331 München, Germany


There is an infinite exchangeable sequence of random variables {Xk}k∈ℕ such that each finitedimensional distribution follows a min-stable multivariate exponential law with Galambos survival copula, named after [7]. A recent result of [15] implies the existence of a unique Bernstein function Ψ associated with {Xk}k∈ℕ via the relation Ψ(d) = exponential rate of the minimum of d members of {Xk}k∈ℕ. The present note provides the Lévy–Khinchin representation for this Bernstein function and explores some of its properties.

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  • [1] S. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 1–66 (1929).

  • [2] L. Bondesson, Classes of infinitely divisible distributions and densities, Z. Wahr. Verw. Geb. 57:1 (1981) pp. 39–71.

  • [3] A. Charpentier, J. Segers, Tails of multivariate Archimedean copulas, J. Multivariate Anal. 100:7 (2009) pp. 1521–1537.

  • [4] B. De Finetti, Funzione caratteristica di un fenomeno allatorio, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 4 (1931) pp. 251–299.

  • [5] B. De Finetti, La prévision: ses lois logiques, ses sources subjectives, Ann. Inst. Henri Poincaré Probab. Stat. 7 (1937) pp. 1–68.

  • [6] K. Es-Sebaiy, Y. Ouknine, How rich is the class of processes which are infinitely divisible with respect to time, Statist. Probab. Lett. 78 (2008) pp. 537–547.

  • [7] J. Galambos, Order statistics of samples from multivariate distributions, J. Amer. Statist. Assoc. 70:351 (1975) pp. 674– 680.

  • [8] G. Gudendorf, J. Segers, Extreme-value copulas, in Copula Theory and Its Applications – Lecture Notes in Statistics, Springer (2010) pp. 127–145.

  • [9] A. Hakassou, Y. Ouknine, A contribution to the study of IDT processes, Working paper, retrievable from http://univi.net/spas/spada2010/tc-ouknine.pdf (2012).

  • [10] F. Hausdorff, Summationsmethoden und Momentfolgen I, Math. Z. 9 (1921) pp. 74–109.

  • [11] F. Hausdorff, Momentenproblem für ein endliches Intervall, Math. Z. 16 (1923) pp. 220–248.

  • [12] E. Hewitt, L.J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955) pp. 470–501.

  • [13] H. Joe, Multivariate models and dependence concepts, Chapman & Hall/CRC (1997).

  • [14] J.-F. Mai, M. Scherer, Lévy-frailty copulas, J. Multivariate Anal. 100 (2009) pp. 1567–1585.

  • [15] J.-F. Mai, M. Scherer, Characterization of extendible distributions with exponential minima via stochastic processes that are infinitely divisible with respect to time, Extremes, in press, DOI 10.1007/s10687-013-0175-4 (2013).

  • [16] R. Mansuy, On processes which are infinitely divisible with respect to time, Working paper, retrievable from http://arxiv.org/abs/math/0504408 (2005).

  • [17] P. Ressel, De Finetti type theorems: an analytical approach, Ann. Probab. 13 (1985) pp. 898–922.

  • [18] K.-I. Sato, Lévy processes and infinitely divisible laws, Cambridge University Press (1999).

  • [19] R. Schilling, R. Song, Z. Vondracek, Bernstein functions, De Gruyter (2010).


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Dependence Modeling aims to provide a medium for exchanging results and ideas in the area of multivariate dependence modeling. Topics include Copula methods, environmental sciences, estimation and goodness-of-fit tests, extreme-value theory, limit laws, mass transportations, measures of association, multivariate distributions and tests, quantitative risk management, risk assessment, risk models, risk measures and stochastic orders and time series.