Moments for self-normalized partial sums

Muneya Matsui, Thomas Mikosch and Olivier Wintenberger

Stochastic Processes and their Applications, 2026, vol. 192, issue C

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.

Keywords: Regularly varying sequence; Sums; Self-normalization; Ratio limits; Moments (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2025.104810

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