Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

Moments of unconditional logarithmically concave vectors

We derive two-sided bounds for moments of linear combinations of coordinates od unconditional log-concave vectors. We also investigate how well moments of such combinations may be approximated by moments of Gaussian random variables.

Fractional smoothness of images of logarithmically concave measures under polynomials

We show that a measure on the real line that is the image of a log-concave measure under a polynomial of degree $d$ possesses a density from the Nikol'skii--Besov class of fractional order $1/d$. This result is used to prove an estimate of the total variation distance between such measures in terms of the Fortet--Mourier distance.

Kantorovich type topologies on spaces of measures and convergence of barycenters

We study two topologies $τ_{KR}$ and $τ_K$ on the space of measures on a completely regular space generated by Kantorovich--Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich--Rubinshtein topology $τ_{KR}$ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A~sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures and stable measures weak convergence implies convergence in the Kantorovich topology. We…

Total variation distance estimates via $L^2$-norm for polynomials in log-concave random vectors

The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.

消化尖锐薄壳不等式的证明

Digesting the proof of the sharp thin-shell inequality

研究高维对数凹分布薄壳定理中通用常数最优值，通过分析加权黎曼流形和蒙日 - 安培方程给出证明，确定\({\rm Var}( |X|^2 ) \leq 8 n\)且常数\(8\)最优，还给出相关张量范数界。