Topological Perspectives on Statistical Quantities I

TL;DR

This paper explores the connection between Boolean cumulants in non-commutative probability and $A_$-algebra theory in algebraic topology, providing a topological perspective on statistical quantities.

Contribution

It introduces a novel link between cumulants in statistics and homotopy theory of algebra maps, bridging two mathematical frameworks.

Findings

Boolean cumulants measure deviation from algebra morphisms

Homotopy theory of $A_$-algebras models non-algebra maps

Establishes a topological interpretation of statistical cumulants

Abstract

In statistics cumulants are defined to be functions that measure the linear independence of random variables. In the non-communicative case the Boolean cumulants can be described as functions that measure deviation of a map between algebras from being an algebra morphism. In Algebraic topology maps that are homotopic to being algebra morphisms are studied using the theory of $A_\infty$ algebras. In this paper we will explore the link between these two points of views on maps between algebras that are not algebra maps.