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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

Indonesian

Dalam kerangka sintesis sistemik yang tepat dari mekanika statistik tertutup, distribusi $q$-Gaussian muncul sebagai solusi entropy maksimum dari entropy Tsallis di bawah pembatasan saat pengawasan yang tepat. Meskipun secara luas digunakan dalam fisika, biologi, geologi, keuangan, dan bidang terkait, distribusi standar $q$-Gaussian secara intrinsik simetris dan tidak dapat menangkap fluktuasi asimetris yang paling sering diamati dalam data empiris. Kami bertanya apakah asimetri dapat muncul dari prinsip entropy maksimum Tsallis tanpa menyimpang secara eksternal atau menyatukan distribusi. Kami menunjukkan bahwa ini dicapai dengan mempertahankan entropy Tsallis dan mendefinisikan rangkaian pengawasan global secara intrinsik dan tidak dapat menangkap fluktuasi asimetris yang berbeda yang sering diamati dalam data empiris.

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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