Recent zbMATH articles in MSC 60B11https://zbmath.org/atom/cc/60B112021-05-28T16:06:00+00:00WerkzeugOn barycenters of probability measures.https://zbmath.org/1459.280132021-05-28T16:06:00+00:00"Berezin, Sergey"https://zbmath.org/authors/?q=ai:berezin.sergey"Miftakhov, Azat"https://zbmath.org/authors/?q=ai:miftakhov.azatLet \(X\) be a Fréchet space and \(\mu\) a Radon probability measure on \(X\). \(a\in X\) is called the barycenter of \(\mu\) if \(\Lambda a=\int_X\Lambda x\mu(dx)\) for any \(\Lambda\in X^*\). Let \(M\) be a non-empty compact convex subset of \(X\). The authors characterize the set of points \(a\in M\) such that \(a\) is the barycenter of some
Radon probability measure \(\mu\) on \(X\) with \(\text{supp }\mu=M\). In particular, if \(X\) is finite-dimensional, this set of barycenters coincides with the relative algebraic interior of \(M\). A counterexample shows that this is not true in infinite-dimensional Hilbert spaces.
Moreover, the authors describe the set of barycenters of measures in the case that \(X=\mathcal{M}(K)\) is the space of signed finite Radon measures on a metric compact space \(K\).
Reviewer: Hans Weber (Udine)Local characteristics and tangency of vector-valued martingales.https://zbmath.org/1459.600912021-05-28T16:06:00+00:00"Yaroslavtsev, Ivan S."https://zbmath.org/authors/?q=ai:yaroslavtsev.ivanSummary: This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic \(L^p\)- and \(\phi \)-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapień, McConnell, and Woyczyński. The vast majority of the assertions presented in the paper is done under the necessary and sufficient UMD assumption on the corresponding Banach space.Squared-norm empirical processes.https://zbmath.org/1459.600542021-05-28T16:06:00+00:00"Vu, Vincent Q."https://zbmath.org/authors/?q=ai:vu.vincent-q"Lei, Jing"https://zbmath.org/authors/?q=ai:lei.jingSummary: This note extends a result of Mendelson on the supremum of a quadratic process to squared norms of functions taking values in a Banach space. Our method of proof is a reduction by a symmetrization argument and simple observation about the additivity of the generic chaining functional. We demonstrate an application to positive linear functionals of the sample covariance matrix and the apparent variance explained by principal components analysis (PCA).