Title: Topological Perspectives on Statistical Quantities I,
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.