Fabian Mies - Empirical Orlicz norms

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Fabian Mies
arXiv, 2510.25408, 2025

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APA Click to copy
Mies, F. (2025). Empirical Orlicz norms. ArXiv, 2510.25408. https://doi.org/10.48550/arXiv.2510.25408

Chicago/Turabian Click to copy
Mies, Fabian. “Empirical Orlicz Norms.” arXiv 2510.25408 (2025).

MLA Click to copy
Mies, Fabian. “Empirical Orlicz Norms.” ArXiv, vol. 2510.25408, 2025, doi:10.48550/arXiv.2510.25408.

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@article{fabian2025a,
  title = {Empirical Orlicz norms},
  year = {2025},
  journal = {arXiv},
  volume = {2510.25408},
  doi = {10.48550/arXiv.2510.25408},
  author = {Mies, Fabian}
}

The empirical Orlicz norm based on a random sample is defined as a natural estimator of the Orlicz norm of a univariate probability distribution. A law of large numbers is derived under minimal assumptions. The latter extends readily to a linear and a nonparametric regression model. Secondly, sufficient conditions for a central limit theorem with a standard rate of convergence are supplied. The conditions for the CLT exclude certain canonical examples, such as the empirical sub-Gaussian norm of normally distributed random variables. For the latter, we discover a nonstandard rate of n^{1/4} \log(n)^{-3/8} with a heavy-tailed, stable limit distribution. It is shown that in general, the empirical Orlicz norm does not admit any uniform rate of convergence for the class of distributions with bounded Orlicz norm.

Keywords: asymptotic distribution, tail inequality, sub-Gaussian random variables, stable distribution