Rafał Latała

Some estimates of norms of random matrices

We show that for any random matrix (X ij ) with independent mean zero entries E∥(X ij )∥ ≤ C(max √ΣEX 2 ij + max √ΣEX 2 ij + 4√ΣEX 4 ij ), where C is some universal constant.

Between sobolev and poincaré

Let a a ∈ [0, 1] and r ∈ [1, 2] satisfy relation r = 2/(2 − a). Let μ(dx)=c r n exp(-(|x1| r +|x2| r +...+|x n | r ))dx1dx2...dx n be a probability measure on the Euclidean space (R n , ‖ · ‖). We prove that there exists a universal constant C such that for any smooth real function f on R n and any p ∈ [1,2)

Estimates of moments and tails of Gaussian chaoses

E_\mu f^2 - (E_\mu \left| f \right|^p )^{2/p} \leqslant C(2 - p)^a E_\mu \left\| {\nabla f} \right\|^2

Small ball probability estimates in terms of width

. We prove also that if for some probabilistic measure μ on R n the above inequality is satisfied for any p ∈ [1, 2) and any smooth f then for any h : R n → R such that |h(x)-h(y)|≤∥x-y∥ there is E μ |h| < ∞ and

A Short Proof of Paouris' Inequality

\mu (h - E_\mu h > \sqrt C \cdot t) \leqslant e^{ - Kt^r }

The LIL for canonical U-statistics of order 2

for t > 1, where K > 0 is some universal constant.