Tomasz Tkocz

A note on a Brunn-Minkowski inequality for the Gaussian measure

We give the counter-examples related to a Gaussian Brunn- Minkowski inequality and the (B) conjecture.

A note on suprema of canonical processes based on random variables with regular moments

We derive two-sided bounds for expected values of suprema of canonical processes based on random variables with moments growing regularly. We also discuss a Sudakov-type minoration principle for canonical processes.

TENSOR PRODUCTS OF RANDOM UNITARY MATRICES

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M = 2, N become large or M become large and N = 2.

Gaussian mixtures: Entropy and geometric inequalities

A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to

Two-sided bounds for Lp-norms of combinations of products of independent random variables

e^{-|t|^p}

Extremal spacings between eigenphases of random unitary matrices and their tensor products

and symmetric

On a Loomis–Whitney Type Inequality for Permutationally Invariant Unconditional Convex Bodies

p

An Upper Bound for Spherical Caps

-stable random variables, where

A note on certain convolution operators

p\in(0,2]

A reverse entropy power inequality for log-concave random vectors

. We obtain various sharp moment and entropy comparison estimates for weighted sums of independent Gaussian mixtures and investigate extensions of the B-inequality and the Gaussian correlation inequality in the context of Gaussian mixtures. We also obtain a correlation inequality for symmetric geodesically convex sets in the unit sphere equipped with the normalized surface area measure. We then apply these results to derive sharp constants in Khintchine inequalities for vectors uniformly distributed on the unit balls with respect to