Homotopical perspective on statistical quantities
Classically, cumulants are functions that measure dependence of random variables. If the random variables are independent then the cumulants are zero. Those notions can be defined in general for linear maps between commutative [\textit{G. C. Drummond-Cole} et al., J. Homotopy Relat. Struct. 10, No. 3, 623--635 (2015; Zbl 1346.55018)] or associative algebras [\textit{F. Lehner}, Math. Z. 248, No. 1, 67--100 (2004; Zbl 1089.46040)] which do not respect the algebraic structure. These cumulants have a close relationship with \(A_\infty\) and \(C_\infty\) morphisms [\textit{J.-L. Loday} and \textit{B. Vallette}, Algebraic operads. Berlin: Springer (2012; Zbl 1260.18001)], which are the classical homotopical tools for analyzing deformations of algebraically compatible linear maps. \par The author studies these two different perspectives to understand how infinity morphisms might help our understanding of cumulants. The main result in this paper is that in the presence of an \(A_\infty\) or \(C_\infty\) morphism, the relevant cumulants are strongly homotopic to zero.
scientific article; zbMATH DE number 7342598
scientific article