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Moments for self-normalized partial sums

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

Abstract

We consider a regularly varyingstationary sequenceof random variables (Xt) with tail index α<2. For this sequence we study the joint convergenceof sums, ℓp-type moduli and maxima. We focus on ratio statistics, including the studentized sums and sums normalized by the corresponding maxima, and study the existence of moments for the limit ratios. We consider particular examples of processes (Xt) whose limit ratios possess all moments as in the iid setting. But, in contrast to the latter situation, there also exist dependent sequences (Xt) where certain moments of the limit ratio are infinite. This phenomenon results from extremal clusters in the sequence.

Suggested Citation

  • Matsui, Muneya & Mikosch, Thomas & Wintenberger, Olivier, 2026. "Moments for self-normalized partial sums," Stochastic Processes and their Applications, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:spapps:v:192:y:2026:i:c:s0304414925002546
    DOI: 10.1016/j.spa.2025.104810
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    References listed on IDEAS

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    4. 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).
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