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Aggregating heavy-tailed random vectors: from finite sums to L\'evy processes

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  • Bikramjit Das
  • Vicky Fasen-Hartmann

Abstract

The tail behavior of aggregates of heavy-tailed random vectors is known to be determined by the so-called principle of "one large jump'', be it for finite sums, random sums, or, L\'evy processes. We establish that, in fact, a more general principle is at play. Assuming that the random vectors are multivariate regularly varying on various subcones of the positive quadrant, first we show that their aggregates are also multivariate regularly varying on these subcones. This allows us to approximate certain tail probabilities which were rendered asymptotically negligible under classical regular variation, despite the "one large jump'' asymptotics. We also discover that depending on the structure of the tail event of concern, the tail behavior of the aggregates may be characterized by more than a single large jump. Eventually, we illustrate a similar phenomenon for multivariate regularly varying L\'evy processes, establishing as well a relationship between multivariate regular variation of a L\'evy process and multivariate regular variation of its L\'evy measure on different subcones.

Suggested Citation

  • Bikramjit Das & Vicky Fasen-Hartmann, 2023. "Aggregating heavy-tailed random vectors: from finite sums to L\'evy processes," Papers 2301.10423, arXiv.org.
  • Handle: RePEc:arx:papers:2301.10423
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    References listed on IDEAS

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    6. Furman, Edward & Kuznetsov, Alexey & Su, Jianxi & Zitikis, Ričardas, 2016. "Tail dependence of the Gaussian copula revisited," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 97-103.
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    Cited by:

    1. 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).
    2. Bikramjit Das & Vicky Fasen-Hartmann, 2023. "On heavy-tailed risks under Gaussian copula: the effects of marginal transformation," Papers 2304.05004, arXiv.org.

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