MathOverflow

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for distribution-valued processes (ProjectEuclid link DOI)for its...

Given a set $A$ and a variety (in the sense of universal algebra) $\mathbb{V}$, let $\mathbb{V}(A)$ be the set of all $\mathbb{V}$-algebras with underlying set $A$. Let $\mathbb{G}$ be the variety of groups. Given a group $G$, I'll write "$\underline...

Say that a class of structures $\mathbb{K}$ (all of the same similarity type) is product-nice iff, whenever $\mathcal{M}_1,...,\mathcal{M}_n\in\mathbb{K}$ and $\varphi(\overline{x})$ is a formula in the language of $\mathbb{K}$, the relation $\prod ...

Let $X$ be a random variable in $\mathbb{R}^d$ with law $\mu$. We denote by $\mathcal{P}(\mathbb{R}^d)$ the set of all Borel probability measures on $\mathbb{R}^d$.
Assume that $\mu \in \mathcal{P}(\mathbb{R}^d)$ has finite second moment. Its mean is$$\bar{\mu} = \mathbb{E}_\mu[X],$$and its variance is$$\sigma^2(\mu) = \mathbb{E}_\mu\big[\|X - \mathbb{E}_\mu[X]\|^2\big].$$...