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The Minkowski length of a spherical random vector

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  • Shushi, Tomer

Abstract

The Rayleigh distribution represents the Euclidean length of a two-dimensional random vector with normally distributed components that are independent, while for the case of a three-dimensional random vector, its length distributed by the well-known Maxwell–Boltzmann distribution. In this letter, we generalize these results in two ways, into the world of elliptical distributions and for general Lp spaces. We present the distribution of the length of an n-dimensional random vector whose components are mutually dependent and symmetrically distributed in Lp spaces. The results show that such distribution has explicit form, which allows computing its moments. Similar to the Rayleigh distribution, the presented distribution can also be useful to model risks. Thus, we derive important risk measures for the investigated distribution.

Suggested Citation

  • Shushi, Tomer, 2019. "The Minkowski length of a spherical random vector," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 104-107.
  • Handle: RePEc:eee:stapro:v:153:y:2019:i:c:p:104-107
    DOI: 10.1016/j.spl.2019.06.003
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    References listed on IDEAS

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    1. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    2. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Tail conditional moments for elliptical and log-elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 179-188.
    3. McNeil, Alexander J. & Frey, Rudiger, 2000. "Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach," Journal of Empirical Finance, Elsevier, vol. 7(3-4), pages 271-300, November.
    4. Shushi, Tomer, 2018. "Stein’s lemma for truncated elliptical random vectors," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 297-303.
    5. Guy Kaplanski & Haim Levy, 2015. "Value-at-risk capital requirement regulation, risk taking and asset allocation: a mean-variance analysis," The European Journal of Finance, Taylor & Francis Journals, vol. 21(3), pages 215-241, February.
    6. 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.
    7. Furman, Edward & Landsman, Zinoviy, 2006. "Tail Variance Premium with Applications for Elliptical Portfolio of Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 433-462, November.
    8. Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
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