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Non-uniform Bounds and Edgeworth Expansions in Self-normalized Limit Theorems

Author

Listed:
  • Pascal Beckedorf

    (University of Freiburg)

  • Angelika Rohde

    (University of Freiburg)

Abstract

We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while imposing only minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized sums.

Suggested Citation

  • Pascal Beckedorf & Angelika Rohde, 2025. "Non-uniform Bounds and Edgeworth Expansions in Self-normalized Limit Theorems," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-94, March.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:1:d:10.1007_s10959-024-01376-8
    DOI: 10.1007/s10959-024-01376-8
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    References listed on IDEAS

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    1. John Robinson & Qiying Wang, 2005. "On the Self-Normalized Cramér-type Large Deviation," Journal of Theoretical Probability, Springer, vol. 18(4), pages 891-909, October.
    2. S. G. Bobkov & G. P. Chistyakov & H. Kösters, 2015. "The Entropic Erdős-Kac Limit Theorem," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1520-1555, December.
    3. Bhattacharya, Rabi N. & Ghosh, J. K., 1988. "On moment conditions for valid formal Edgeworth expansions," Journal of Multivariate Analysis, Elsevier, vol. 27(1), pages 68-79, October.
    4. Finner, Helmut & Roters, Markus & Dickhaus, Thorsten, 2007. "Characterizing Density Crossing Points," The American Statistician, American Statistical Association, vol. 61, pages 28-33, February.
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