A formula for hidden regular variation behavior for symmetric stable distributions
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A formula for hidden regular variation behavior for symmetric stable distributions ¨ 1,2 Malin Palo¨ Forsstrom
· Jeffrey E. Steif1
Received: 22 October 2019 / Revised: 10 March 2020 / Accepted: 27 May 2020 / © The Author(s) 2020
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. Keywords Hidden regular variation · Multivariate stable distributions AMS 2000 Subject Classifications 60E07 · 60G70
1 Main result and remarks Many distributions have tails that exhibit regular variation (see Bingham et al. (1987) and Mikosch and Wintenberger (2019)) which means that they behave like a power-law times a slowly varying function. Examples are one-dimensional stable distributions where the slowly varying function is just constant. For stable random vectors, one also has this but in addition, one can have more interesting behavior, so-called hidden regular variation (see Das et al. (2013), Resnick (2002), Resnick 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. Malin Pal¨o Forsstr¨om
[email protected] Jeffrey E. Steif [email protected] 1
Chalmers University of Technology and Gothenburg University, Gothenburg, Sweden
2
KTH Royal Institute of Technology, Stockholm, Sweden
M.P. Forsstr¨om, J.E. Steif
(2007)), meaning that one has different power-law decay in different directions. Ideally, one would like to capture the correct decay rate in each such direction. Our main result, Theorem 1, describes such behavior for symmetric stable distributions. Needed definitions and background will be given in Section 2. Let α ∈ (0, 2) and X be an n-dimensional symmetric α-stable random vector with spectral measure Λ (see Eq. 4). Then Λ is a bounded measure on the unit sphere ¯ where E¯ and E o denote the Sn−1 in Rn . Let E ⊆ Rn be a Borel set with 0 ∈ E, closure and interior of E respectively. With d being the Euclidean distance, define the δ-neighborhood of E by Eδ,+ := {x ∈ Rn : d(x, E) < δ}. For any integer k ≥ 1 and any E as above, letting Cα be a constant defined in Section 2 and Λk := Λ × · · · × Λ (k times), define ∞ Ck ∞ L(E, k, α) := α ··· k! 0 0 k k −(1+α) ×Λk x1 , . . . , xk ∈ Sn−1 : si xi ∈ E · αsi dsi . (1) i=1
i=1
Theorem 1 For any α, X, E and k as above, L(E o , k, α) ≤ lim inf hkα P (X ∈ hE) h→∞
≤ lim sup hkα P (X ∈ hE) h→∞
≤ lim L(Eδ,+ , k, α). δ→0
(2)
¯ k, α) ≤ limδ→0 L(Eδ,+ , k, α). Remark 1 1. Clearly L(E o , k, α) ≤ L(E, ¯ k, α) = Also, by the Lebesgue dominated convergence theorem, L(E, limδ→0 L(Eδ,+
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