Rademacher inequalities with applications
Let \(\varepsilon,\varepsilon_1,\varepsilon_2,\dots\) be a sequence of i.i.d.\ Rademacher random variables with \(\operatorname{P}(\varepsilon=\pm1)=1/2\). Let \(X_1,\tilde{X_1},X_2,\tilde{X_2},\dots\) be independent, non-negative random variables with \(\operatorname{E}f(X_i)\geq \operatorname{E}f(\tilde{X_i})\) for all \(i\) and all convex functions \(f\). The authors begin by finding those \(p\) and \(q>0\) for which either \[ \operatorname{E}|X_1^p\varepsilon_1+\dots+X_n^p\varepsilon_n|^q\geq \operatorname{E}|\tilde{X_1}^p\varepsilon_1+\dots+\tilde{X_n}^p\varepsilon_n|^q, \] or the reversed inequality holds. Such inequalities are then extended to general Hilbert spaces by replacing the Rademacher variables with independent Hilbert space valued symmetric random vectors. Applications are given to a trace norm of symmetric random matrices. Further results and applications are also given for analogous comparisons of random matrices and of freely independent random variables.
scientific article; zbMATH DE number 6348425
scientific article