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A note on the Galambos copula and its associated Bernstein function

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Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
1
Opis fizyczny
Daty
otrzymano
2013-12-04
zaakceptowano
2014-02-19
online
2014-03-20
Twórcy
  • Lehrstuhl für Finanzmathematik (M13), Technische Universität München, Parkring 11,
    85748 Garching-Hochbrück, Germany, janfrederik.mai@xaia.com,
  • XAIA Investment GmbH, Sonnenstraße 19, 80331 München, Germany
Bibliografia
  • [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.[WoS][Crossref]
  • [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.[WoS][Crossref]
  • [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 fromhttp://univi.net/spas/spada2010/tc-ouknine.pdf (2012).
  • [10] F. Hausdorff, Summationsmethoden und Momentfolgen I, Math. Z. 9 (1921) pp. 74–109.[Crossref]
  • [11] F. Hausdorff, Momentenproblem für ein endliches Intervall, Math. Z. 16 (1923) pp. 220–248.[Crossref]
  • [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.[Crossref]
  • [15] J.-F. Mai, M. Scherer, Characterization of extendible distributions with exponential minima via stochastic processes thatare infinitely divisible with respect to time, Extremes, in press, DOI 10.1007/s10687-013-0175-4 (2013).[Crossref]
  • [16] R. Mansuy, On processes which are infinitely divisible with respect to time, Working paper, retrievable fromhttp://arxiv.org/abs/math/0504408 (2005).
  • [17] P. Ressel, De Finetti type theorems: an analytical approach, Ann. Probab. 13 (1985) pp. 898–922.[Crossref]
  • [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).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_demo-2014-0002
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