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Rafał Latała

Publications and source records attributed to Rafał Latała.

46 records · Page 3Linked to original sources
arXiv2008

On the infimum convolution inequality

In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC-inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure. In particular, we show the optimal IC-inequality for product log-concave measures and for uniform measures on the l_p^n balls. Such an optimal inequality implies, for a given measure, in particular the Central Limit Theorem of Klartag and the tail estimates of Paouris.

math.PR↗
arXiv2007

On Weak Tail Domination of Random Vectors

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture.

math.PR↗
arXiv2007

The LIL for $U$-statistics in Hilbert spaces

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for $U$-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued $U$-statistics of arbitrary order, which are of independent interest.

math.PR↗
arXiv2007

Estimates of moments and tails of Gaussian chaoses

We derive two-sided estimates on moments and tails of Gaussian chaoses, that is, random variables of the form $\sum a_{i_1,...,i_d}g_{i_1}... g_{i_d}$, where $g_i$ are i.i.d. ${\mathcal{N}}(0,1)$ r.v.'s. Estimates are exact up to constants depending on $d$ only.

math.PR↗
arXiv2005

Small ball probability estimates in terms of width

A certain inequality conjectured by Vershynin is studied. It is proved that for any $n$-dimensional symmetric convex body $K$ with inradius $w$ and $γ_{n}(K) \leq 1/2$ there is $γ_{n}(sK) \leq (2s)^{w^{2}/4}γ_{n}(K)$ for any $s \in [0,1]$. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.

math.PR↗
arXiv2003

On some inequalities for Gaussian measures

We review several inequalities concerning Gaussian measures - isoperimetric inequality, Ehrhard's inequality, Bobkov's inequality, S-inequality and correlation conjecture.

math.PR↗
arXiv2000

Exponential and moment inequalities for U-statistics

A Bernstein-type exponential inequality for (generalized) canonical U-statistics of order 2 is obtained and the Rosenthal and Hoffmann-Jørgensen inequalities for sums of independent random variables are extended to (generalized) U-statistics of any order whose kernels are either nonnegative or canonical

math.PR↗
arXiv2000

Between Sobolev and Poincaré

We establish a family of functional inequalities interpolating between the classical logarithmic Sobolev and Poincaré inequalities. We prove that they imply the concentration of measure phenomenon intermediate between Gaussian and exponential. Our bounds are close to optimal.

math.PR↗
arXiv1999

The LIL for canonical U-statistics of order 2

Let X,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\limsup_n (n\log\log n)^{-1}|\sum_{1<= i< j<= n}h(X_i,X_j)|<\infty$ a.s., holds if and only if the following three conditions are satisfied: h is canonical for the law of X (that is Eh(X,y)=0 for almost y) and there exists $C<\infty$ such that, both, $E\min(h^2(X_1,X_2),u)<C\log\log u$ for all large u and $sup\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X)|_2<1,\|g(X)\|_2<1, \|f\|_\infty<\infty, \|g\|_\infty<\infty\}< C$.

math.PR↗