Coercive Inequalities and U-Bounds on Step-Two Carnot Groups

We prove Poincaré and Logβ-Sobolev inequalities for a class of probability measures on step-two Carnot groups.

Coercive inequalities in higher-dimensional anisotropic heisenberg group

In the setting of higher-dimensional anisotropic Heisenberg group, we compute the fundamental solution for the sub-Laplacian, and we prove Poincaré and $$\beta $$ β -Logarithmic Sobolev inequalities for measures as a function of this fundamental solution.

Concentration Inequalities on the Multislice and for Sampling Without Replacement

We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application, we show concentration results for the triangle count in the G(n, M) Erdős–Rényi model resembling known bounds in the G(n, p) case. Moreover, we give a proof of Talagrand’s convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for n out of N sampling without replacement. Based on a bounded difference inequality involving the finite-sampling correction factor $$1 - (n / N)$$ 1 - ( n / N ) , we present an easy proof of Serfling’s inequality with a slightly worse factor in the exponent, as well as a sub-Gaussian right tail for the Kolmogorov distance between the empirical measure and the true distribution of the sample.