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On the regular variation of elliptical random vectors

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  • Hashorva, Enkelejd

Abstract

Let S=(S1,...,Sd)[inverted perpendicular],d[greater-or-equal, slanted]2 be a spherical random vector in and let X=A[inverted perpendicular]S be an elliptical random vector with a non-singular matrix. Berman (1992. Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole) proved that if the random radius is regularly varying with index [alpha]>0 then S and Si,1[less-than-or-equals, slant]i[less-than-or-equals, slant]d are regularly varying with index [alpha]. In this paper we derive several new equivalent conditions for the regular variation of X. As a by-product we obtain two asymptotic results concerning the sojourn and supremum of Berman processes.

Suggested Citation

  • Hashorva, Enkelejd, 2006. "On the regular variation of elliptical random vectors," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1427-1434, August.
    • Handle: RePEc:eee:stapro:v:76:y:2006:i:14:p:1427-1434
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    References listed on IDEAS

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    1. Thomas Mikosch, 2005. "How to model multivariate extremes if one must?," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 59(3), pages 324-338, August.
    2. Kotz, Samuel & Nadarajah, Saralees, 2001. "Some extremal type elliptical distributions," Statistics & Probability Letters, Elsevier, vol. 54(2), pages 171-182, September.
    3. Hashorva, Enkelejd, 2005. "Extremes of asymptotically spherical and elliptical random vectors," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 285-302, June.
    4. Kano, Y., 1994. "Consistency Property of Elliptic Probability Density Functions," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 139-147, October.
    5. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
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    Cited by:

    1. Cai, J., 2012. "Estimation concerning risk under extreme value conditions," Other publications TiSEM a92b089f-bc4c-41c2-b297-c, Tilburg University, School of Economics and Management.
    2. Opitz, T., 2013. "Extremal t processes: Elliptical domain of attraction and a spectral representation," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 409-413.
    3. Hashorva, Enkelejd, 2010. "On the residual dependence index of elliptical distributions," Statistics & Probability Letters, Elsevier, vol. 80(13-14), pages 1070-1078, July.
    4. Cai, J. & Einmahl, J.H.J. & de Haan, L.F.M., 2011. "Estimation of extreme risk regions under multivariate regular variation," Other publications TiSEM b7a72a8d-f9bc-4129-ae9b-a, Tilburg University, School of Economics and Management.
    5. Yves Dominicy & Pauliina Ilmonen & David Veredas, 2017. "Multivariate Hill Estimators," International Statistical Review, International Statistical Institute, vol. 85(1), pages 108-142, April.
    6. Constantinescu, Corina & Hashorva, Enkelejd & Ji, Lanpeng, 2011. "Archimedean copulas in finite and infinite dimensions—with application to ruin problems," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 487-495.
    7. Hashorva, Enkelejd, 2010. "Asymptotics of the norm of elliptical random vectors," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 926-935, April.
    8. Hashorva, Enkelejd, 2007. "Extremes of conditioned elliptical random vectors," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1583-1591, September.

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