Abstract
We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in “different directions”, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.
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Acknowledgements
We thank Gennady Samorodnitsky for private correspondence and for elaborating on one of the proofs in Samorodnitsky and Taqqu (1994). We also thank Thomas Mikosch for informing us about the notion of hidden regular variation and its relationship to our work and two anonymous referees for various useful comments.
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The first author acknowledges support from the European Research Council, grant no. 682537. The second author acknowledges the support of the Swedish Research Council, grant no. 2016-03835 and the Knut and Alice Wallenberg Foundation, grant no. 2012.0067.
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Forsström, M.P., Steif, J.E. A formula for hidden regular variation behavior for symmetric stable distributions. Extremes 23, 667–691 (2020). https://doi.org/10.1007/s10687-020-00381-4
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DOI: https://doi.org/10.1007/s10687-020-00381-4