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On regular variation of probability densities

Author

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  • de Haan, L.
  • Resnick, S.

Abstract

Regular variation of the tail of a multivariate probability distribution is implied by regular variation of the density f provided f satisfies a regularity condition. We give a uniformity condition which controls variation of the function f across rays. Our condition is somewhat more flexible than the usual regularity condition of monotonicity. Some examples are given. As a by-product we get results on multidimensional regular variation of some independent interest.

Suggested Citation

  • de Haan, L. & Resnick, S., 1987. "On regular variation of probability densities," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 83-93.
    • Handle: RePEc:eee:spapps:v:25:y:1987:i::p:83-93
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    Citations

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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. Li, Haijun & Hua, Lei, 2015. "Higher order tail densities of copulas and hidden regular variation," Journal of Multivariate Analysis, Elsevier, vol. 138(C), pages 143-155.
    3. Das, Bikramjit & Fasen-Hartmann, Vicky, 2024. "On heavy-tailed risks under Gaussian copula: The effects of marginal transformation," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    4. Yun, Seokhoon, 1997. "On Domains of Attraction of Multivariate Extreme Value Distributions under Absolute Continuity," Journal of Multivariate Analysis, Elsevier, vol. 63(2), pages 277-295, November.
    5. Yi He & John H. J. Einmahl, 2017. "Estimation of extreme depth-based quantile regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 449-461, March.
    6. Anand Deo & Karthyek Murthy, 2020. "Optimizing tail risks using an importance sampling based extrapolation for heavy-tailed objectives," Papers 2008.09818, arXiv.org.
    7. Omey, Edward & Van Gulck, Stefan, 2015. "Intuitive approximations in discrete renewal theory, Part 1: Regularly varying case," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 68-74.
    8. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    9. 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.
    10. Bikramjit Das & Tiandong Wang & Gengling Dai, 2022. "Asymptotic Behavior of Common Connections in Sparse Random Networks," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2071-2092, September.
    11. Tiandong Wang & Sidney I. Resnick, 2018. "Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 1029-1042, September.

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