Standard deviation is a strongly Leibniz seminorm

TL;DR

This paper demonstrates that standard deviation functions as a strongly Leibniz seminorm, satisfying specific inequalities in both classical and non-commutative probability spaces, with extensions to C*-algebras and conditional expectations.

Contribution

It establishes that standard deviation is a strongly Leibniz seminorm and extends this property to non-commutative probability spaces and C*-algebra contexts.

Findings

Standard deviation satisfies Leibniz inequality for bounded functions.

The inequalities hold in non-commutative probability spaces.

Extensions to matricial seminorms and conditional expectations are provided.

Abstract

We show that standard deviation $\s$ satisfies the Leibniz inequality $\s(fg) \leq \s(f)\|g\| + \|f\|\s(g)$ for bounded functions f, g on a probability space, where the norm is the supremum norm. A related inequality that we refer to as "strong" is also shown to hold. We show that these in fact hold also for non-commutative probability spaces. We extend this to the case of matricial seminorms on a unital C*-algebra, which leads us to treat also the case of a conditional expectation from a unital C*-algebra onto a unital C*-subalgebra.