Statistics of the Shepp p-product
Abstract
In 1962, Shepp famously discovered a product of normal random variables that preserves normality. The Shepp product, which takes the form XY/√(X2 + Y2), has since been thoroughly studied and has found numerous connections to other areas of statistics. Among other things, it has an extension to n normal variables, gives a multiplicative analogue of the central limit theorem, and applies rather unexpectedly to genomics as a test statistic for alignment-free sequence analysis. The Shepp product is evidently the p =2 special case of XY/√[p]Xp + Yp that we call the Shepp p-product. We will show that the Shepp p-product, particularly when p = 1 and p = ∞ (the latter in a limiting sense), is no less fascinating and applicable than the original p = 2 case. Just as the Shepp $2$-product preserves normal distributions, the Shepp $1$-product preserves Cauchy distributions, and the Shepp ∞-product preserves exponential distributions. In fact, the converse is also true in an appropriate sense, allowing us to characterize the Cauchy, normal, and exponential distributions as the unique distributions preserved by the Shepp p-product for p =1,2,∞, respectively. We will study the multiplicative analogue of infinite divisibility with respect to the Shepp p-product, establish an asymptotic theory for the Shepp p-product of n i.i.d.\ random variables, and estimate the rates of convergence in Kolmogorov distance. Some of our results are new even for the p=2 case. We will also discuss new applications of the Shepp p-product in statistics, computational biology, and statistical physics. In particular, our new test statistic for exponentiality based on the Shepp ∞-product is more powerful, both theoretically and empirically, than the rule-of-thumb Kolmogorov--Smirnov test based on the relative efficiency.
Related
- The higher-dimensional Shepp problem: an exact criterion for random ball coverings of tori
- The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem
- Minimum-Time Escape from the Conic Field of View of a Reeds--Shepp Pursuer: the Faster Evader
- Minimum-Time Escape from the Conic Field of View of a Reeds-Shepp Pursuer: Classification and Game Value for a Slower or Equally Fast Evader
- Limit theory for Lipschitz-localized statistics in random geometric models
BibTeX
@misc{guo2026statistics,
title = {{Statistics of the Shepp $p$-product}},
author = {Wenxuan Guo and Lek-Heng Lim},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03282},
eprint = {2611.03282},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03282},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Statistics of the Shepp p-product
AU - Wenxuan Guo
AU - Lek-Heng Lim
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03282
UR - https://arxiv.org/abs/2611.03282
AB - In 1962, Shepp famously discovered a product of normal random variables that preserves normality. The Shepp product, which takes the form XY/√(X^2 + Y^2), has since been thoroughly studied and has found numerous connections to other areas of statistics. Among other things, it has an extension to n normal variables, gives a multiplicative analogue of the central limit theorem, and applies rather unexpectedly to genomics as a test statistic for alignment-free sequence analysis. The Shepp product is evidently the p =2 special case of XY/√[p]X^p + Y^p that we call the Shepp p-product. We will show that the Shepp p-product, particularly when p = 1 and p = ∞ (the latter in a limiting sense), is no less fascinating and applicable than the original p = 2 case. Just as the Shepp $2$-product preserves normal distributions, the Shepp $1$-product preserves Cauchy distributions, and the Shepp ∞-product preserves exponential distributions. In fact, the converse is also true in an appropriate sense, allowing us to characterize the Cauchy, normal, and exponential distributions as the unique distributions preserved by the Shepp p-product for p =1,2,∞, respectively. We will study the multiplicative analogue of infinite divisibility with respect to the Shepp p-product, establish an asymptotic theory for the Shepp p-product of n i.i.d.\ random variables, and estimate the rates of convergence in Kolmogorov distance. Some of our results are new even for the p=2 case. We will also discuss new applications of the Shepp p-product in statistics, computational biology, and statistical physics. In particular, our new test statistic for exponentiality based on the Shepp ∞-product is more powerful, both theoretically and empirically, than the rule-of-thumb Kolmogorov--Smirnov test based on the relative efficiency.
ER -