Classical cumulants from the double tensor unshuffle Hopf algebra

Xing Gao; Jingtao Li

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arXiv · 2610.33382

Classical cumulants from the double tensor unshuffle Hopf algebra

Xing Gao·Jingtao Li

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Abstract

We realize classical cumulants and their fixed-point equations in the double tensor unshuffle Hopf algebra introduced by Ebrahimi-Fard and Patras, placing classical, free, Boolean, and monotone cumulants in a common algebraic framework. A concatenation perturbation of the functionals yields left and right half-shuffle equations on ordinary words without changing the coproduct or either half-coproduct. Both equations are equivalent to the classical moment--cumulant formula and uniquely determine the classical cumulants. We also obtain a Hopf-algebraic realization of the classical-to-free cumulant transform through convolution exponentiation and the left half-shuffle logarithm. A commutative diagram identifies its partition-level action with summation over non-crossing closure fibers, recovering Lehner's closure-fiber formula within the framework of the double tensor unshuffle Hopf algebra.

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Xing Gao, Jingtao Li. 2026-10-24. Classical cumulants from the double tensor unshuffle Hopf algebra. https://arxiv.org/abs/2610.33382

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