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Self-normalized partial sums of heavy-tailed time series

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  • Matsui, Muneya
  • Mikosch, Thomas
  • Wintenberger, Olivier

Abstract

We study the joint limit behavior of sums, maxima and ℓp-type moduli for samples taken from an Rd-valued regularly varying stationary sequence with infinite variance. As a consequence, we can determine the distributional limits for ratios of sums and maxima, studentized sums, and other self-normalized quantities in terms of hybrid characteristic-distribution functions and Laplace transforms. These transforms enable one to calculate moments of the limits and to characterize the differences between the iid and stationary cases in terms of indices which describe effects of extremal clustering on functionals acting on the dependent sequence.

Suggested Citation

  • Matsui, Muneya & Mikosch, Thomas & Wintenberger, Olivier, 2025. "Self-normalized partial sums of heavy-tailed time series," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:spapps:v:190:y:2025:i:c:s030441492500170x
    DOI: 10.1016/j.spa.2025.104729
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    References listed on IDEAS

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    1. Janßen, Anja, 2019. "Spectral tail processes and max-stable approximations of multivariate regularly varying time series," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1993-2009.
    2. Albrecher, Hansjörg & García Flores, Brandon, 2022. "Asymptotic analysis of generalized Greenwood statistics for very heavy tails," Statistics & Probability Letters, Elsevier, vol. 185(C).
    3. Kulik, Rafał & Soulier, Philippe & Wintenberger, Olivier, 2019. "The tail empirical process of regularly varying functions of geometrically ergodic Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4209-4238.
    4. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
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    1. Matsui, Muneya & Mikosch, Thomas & Wintenberger, Olivier, 2026. "Moments for self-normalized partial sums," Stochastic Processes and their Applications, Elsevier, vol. 192(C).

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