### A Berry-Esseen theorem on semisimple Lie groups

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The inequality $\u0283log|\sum {a}_{n}{e}^{2\pi i{\phi}_{n}}|d{\phi}_{1}\dots d{\phi}_{n}\ge Clog\left(\sum \right|{a}_{n}{{|}^{2})}^{1/2}$ with an absolute constant C, and similar ones, are extended to the case of ${a}_{n}$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by ${e}^{2\pi i\phi}$.

By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived...

We prove a kind of logarithmic Sobolev inequality claiming that the mutual free Fisher information dominates the microstate free entropy adapted to projections in the case of two projections.

We consider probability measures supported on a finite discrete interval [0, n]. We introduce a new finite difference operator ∇n, defined as a linear combination of left and right finite differences. We show that this operator ∇n plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport...

In this paper, we present a new proof of the celebrated theorem of Kellerer, stating that every integrable process, which increases in the convex order, has the same one-dimensional marginals as a martingale. Our proof proceeds by approximations, and calls upon martingales constructed as solutions of stochastic differential equations. It relies on a uniqueness result, due to Pierre, for a Fokker-Planck equation.