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Asymmetric $q$-Gaussian Distributions from Tsallis Maximum Entropy

Francesco Cesarone, Massimiliano Corradini, Nicolò Giunta, Serena Rocci · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.16663 · cited by 0 works (OpenAlex) · open-access PDF

Korean

Finricricricricity는 특정 폐쇄 통계 기계의 프레임 워크 내에서 $q$-Gaussian 배포가 적절한 배포 순간 제한에 따라 Tsallis 엔트로피의 최대 엔트로피 솔루션으로 나타납니다. Finricricricity는 물리학, 생물학, 지질학, 재무 및 관련 분야에서 널리 사용되지만, 표준 $q$-Gaussian은 본질적으로 대칭적이며 경험적 데이터에서 종종 관찰되는 대부분의 비대칭 변동을 캡처할 수 없습니다. 우리는 비대칭이 외부로 구부러지거나 배포에 합류하지 않고 Tsallis 최대 엔트로피 원리를 반환 할 수 있는지 여부를 묻습니다. 우리는 표준 Tsallis 엔트로피를 유지하여 표준 $q$-Gaus

Machine translation (Meta M2M100); the English original follows.

Abstract

Within the framework of non-extensive statistical mechanics, $q$-Gaussian distributions arise as maximum-entropy solutions of Tsallis entropy under suitable escort moment constraints. Although widely used in physics, biology, geology, finance, and related fields, standard $q$-Gaussians are intrinsically symmetric and cannot capture asymmetric fluctuations often observed in empirical data. We ask whether asymmetry can arise from the Tsallis maximum-entropy principle without externally skewing or joining distributions. We show that this is achieved by retaining the Tsallis entropy and imposing a global escort mean together with distinct downside and upside escort semi-variances. The key step is to reformulate the problem of maximum Tsallis entropy in terms of the escort density. The nonlinear escort constraints then become affine, while maximizing Tsallis entropy becomes equivalent to maximizing a strictly concave functional over a convex feasible set. Consequently, any admissible stationary solution is the unique global maximizer of the corresponding variational problem. The finite-sample approximation properties of the proposed asymmetric $q$-Gaussian are studied through Monte Carlo experiments using synthetic data generated from Azzalini skew-Student distributions. Compared with the classical symmetric $q$-Gaussian, the asymmetric specification systematically provides a closer approximation to synthetic asymmetric heavy-tailed distributions. Finally, we fit the model to return series from major equity indices, a commodity, and cryptocurrencies, and compare its in-sample goodness of fit with that of a symmetric $q$-Gaussian. The results show that the asymmetric model offers a closer in-sample description of unconditional return distributions, most clearly in the tails.

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Topics: Tsallis entropy, Mathematics, Statistical physics, Asymmetry, Entropy (arrow of time), Nonlinear system

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