Does the Power-Law Advantage in the Tails Outweigh the Global q-Gaussian Description?
Abstract
Financial markets are complex systems whose return distributions exhibit heavy tails, traditionally described by power laws. An alternative framework based on Tsallis nonextensive statistical mechanics allows both the central region and the tails of the distribution to be represented by a single q-Gaussian functional form. In this work, we systematically compare these two descriptions across different time scales, considering stock markets from Brazil and the United States, as well as traditional assets and cryptocurrencies. The q-Gaussian parameters are estimated using the method of moments ratio, whereas the power-law parameters are obtained by maximum likelihood estimation in the asymptotic region. The quality of the fits is assessed using goodness-of-fit measures, bootstrap resampling, and the Vuong test. When the analysis is restricted exclusively to the extreme tails, the power law generally provides a better description, as expected from the asymptotic power-law behavior of the q-Gaussian itself. However, this superiority is not sufficiently pronounced to outweigh the main advantage of the q-Gaussian: its ability to provide a consistent description of the entire distribution, including the central peak, bulk, and tails, over all investigated time scales. Moreover, the evolution of the parameter q offers a simple and direct characterization of aggregational Gaussianity and of the transition between heavy- and short-tailed statistical regimes, without requiring separate fits for different regions of the distribution. These results indicate that, although the power law remains particularly suitable for describing extreme events, the q-Gaussian provides a broader and more practical framework for characterizing the statistical evolution of financial returns.
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BibTeX
@misc{boor2026does,
title = {{Does the Power-Law Advantage in the Tails Outweigh the Global q-Gaussian Description?}},
author = {Eduardo Boor and Roberto da Silva and Joao Carlos Schmitt de Siqueira and J. Roberto Iglesias},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30219},
eprint = {2608.30219},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30219},
}
RIS
TY - UNPB
TI - Does the Power-Law Advantage in the Tails Outweigh the Global q-Gaussian Description?
AU - Eduardo Boor
AU - Roberto da Silva
AU - Joao Carlos Schmitt de Siqueira
AU - J. Roberto Iglesias
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30219
UR - https://arxiv.org/abs/2608.30219
AB - Financial markets are complex systems whose return distributions exhibit heavy tails, traditionally described by power laws. An alternative framework based on Tsallis nonextensive statistical mechanics allows both the central region and the tails of the distribution to be represented by a single q-Gaussian functional form. In this work, we systematically compare these two descriptions across different time scales, considering stock markets from Brazil and the United States, as well as traditional assets and cryptocurrencies. The q-Gaussian parameters are estimated using the method of moments ratio, whereas the power-law parameters are obtained by maximum likelihood estimation in the asymptotic region. The quality of the fits is assessed using goodness-of-fit measures, bootstrap resampling, and the Vuong test. When the analysis is restricted exclusively to the extreme tails, the power law generally provides a better description, as expected from the asymptotic power-law behavior of the q-Gaussian itself. However, this superiority is not sufficiently pronounced to outweigh the main advantage of the q-Gaussian: its ability to provide a consistent description of the entire distribution, including the central peak, bulk, and tails, over all investigated time scales. Moreover, the evolution of the parameter q offers a simple and direct characterization of aggregational Gaussianity and of the transition between heavy- and short-tailed statistical regimes, without requiring separate fits for different regions of the distribution. These results indicate that, although the power law remains particularly suitable for describing extreme events, the q-Gaussian provides a broader and more practical framework for characterizing the statistical evolution of financial returns.
ER -