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Sep 21, 2013
09/13

by
Rafał Latała

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We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.

Source: http://arxiv.org/abs/1104.0372v1

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50

Sep 17, 2013
09/13

by
Rafał Latała

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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.

Source: http://arxiv.org/abs/0711.1477v1

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Sep 21, 2013
09/13

by
Rafał Latała

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We establish upper bounds for tails of order statistics of isotropic log-concave vectors and apply them to derive a concentration of l_r norms of such vectors.

Source: http://arxiv.org/abs/1011.6610v1

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Jun 25, 2018
06/18

by
Rafał Latała

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Let $X_1,X_2,\ldots,X_n$ be independent random variables and $S_k=\sum_{i=1}^k X_i$. We show that for any constants $a_k$, \[ \Pr(\max_{1\leq k\leq n}||S_{k}|-a_{k}|>11t)\leq 30 \max_{1\leq k\leq n}\Pr(||S_{k}|-a_{k}|>t). \] We also discuss similar inequalities for sums of Hilbert and Banach space valued random vectors.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1501.00698

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Sep 21, 2013
09/13

by
Rafał Latała

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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.

Source: http://arxiv.org/abs/1104.0373v1

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Sep 23, 2013
09/13

by
Rafał Latała

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We discuss a conjecture about comparability of weak and strong moments of log-concave random vectors and show the conjectured inequality for unconditional vectors in normed spaces with a bounded cotype constant.

Source: http://arxiv.org/abs/1012.2703v1

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Sep 19, 2013
09/13

by
Rafał Latała

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

Source: http://arxiv.org/abs/math/0304343v1

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Sep 18, 2013
09/13

by
Rafał Latała

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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.

Source: http://arxiv.org/abs/math/0505313v2

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Jul 20, 2013
07/13

by
Radosław Adamczak; Rafał Latała

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We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.

Source: http://arxiv.org/abs/1007.1431v1

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3.0

Jun 30, 2018
06/18

by
Rafał Latała; Tomasz Tkocz

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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.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1406.6584

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3.0

Jun 29, 2018
06/18

by
Rafał Latała; Marta Strzelecka

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We show that for $p\ge 1$, the $p$-th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak $p$-th moment provided that $2q$-th and $q$-th integral moments of these variables are comparable for all $ q \ge 2$. The latest condition turns out to be necessary in the i.i.d. case.

Topics: Probability, Metric Geometry, Mathematics

Source: http://arxiv.org/abs/1612.02407

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Sep 22, 2013
09/13

by
Rafał Latała; Krzysztof Oleszkiewicz

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

Source: http://arxiv.org/abs/math/0003043v1

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Jun 26, 2018
06/18

by
Rafał Latała; Marta Strzelecka

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We show that for $p\geq 1$ and $r\geq 1$ the $p$-th moment of the $l_r$-norm of a log-concave random vector is comparable to the sum of the first moment and the weak $p$-th moment up to a constant proportional to $r$. This extends the previous result of Paouris concerning Euclidean norms.

Topics: Probability, Metric Geometry, Mathematics

Source: http://arxiv.org/abs/1501.01649

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Jun 28, 2018
06/18

by
Konrad Kolesko; Rafał Latała

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We derive two-sided estimates for random multilinear forms (random chaoses) generated by independent symmetric random variables with logarithmically concave tails. Estimates are exact up to multiplicative constants depending only on the order of chaos.

Topics: Mathematics, Probability

Source: http://arxiv.org/abs/1508.07541

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Sep 22, 2013
09/13

by
Rafał Latała; Krzysztof Oleszkiewicz

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

Source: http://arxiv.org/abs/math/0501268v1

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Jul 20, 2013
07/13

by
Radosław Adamczak; Rafał Latała

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We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for canonical $U$-statistics of arbitrary order $d$, extending the previously known results for $d=2$. The nasc's are expressed as growth conditions on a parameterized family of norms associated with the $U$-statistics kernel.

Source: http://arxiv.org/abs/math/0604262v2

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Sep 18, 2013
09/13

by
Rafał Latała; Joel Zinn

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Under some mild regularity on the normalizing sequence, we obtain necessary and sufficient conditions for the Strong Law of Large Numbers for (symmetrized) U-statistics. We also obtain nasc's for the a.s. convergence of series of an analogous form.

Source: http://arxiv.org/abs/math/9901068v1

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Sep 18, 2013
09/13

by
Radosław Adamczak; Rafał Latała

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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.

Source: http://arxiv.org/abs/0704.1643v1

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Sep 23, 2013
09/13

by
Rafał Latała; Jakub Onufry Wojtaszczyk

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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...

Source: http://arxiv.org/abs/0801.4036v1

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Sep 22, 2013
09/13

by
Evarist Giné; Rafał Latała; Joel Zinn

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

Source: http://arxiv.org/abs/math/0003228v1

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4.0

Jun 30, 2018
06/18

by
Radosław Adamczak; Rafał Latała; Zbigniew Puchała; Karol Życzkowski

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We analyze entropic uncertainty relations for two orthogonal measurements on a $N$-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix $U$ relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than these obtained with use of the entropic uncertainty relation by Maassen and Uffink,...

Topics: Quantum Physics, Mathematics, Computing Research Repository, Information Theory, Probability,...

Source: http://arxiv.org/abs/1412.7065

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3.0

Jun 30, 2018
06/18

by
Ewa Damek; Rafał Latała; Piotr Nayar; Tomasz Tkocz

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We show that for every positive p, the L_p-norm of linear combinations (with scalar or vector coefficients) of products of i.i.d. random variables, whose moduli have a nondegenerate distribution with the p-norm one, is comparable to the l_p-norm of the coefficients and the constants are explicit. As a result the same holds for linear combinations of Riesz products. We also establish the upper and lower bounds of the L_p-moments of partial sums of perpetuities.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1404.0344

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39

Sep 21, 2013
09/13

by
Evarist Giné; Stanisław Kwapień; Rafał Latała; Joel Zinn

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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

Source: http://arxiv.org/abs/math/9905132v1

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Jul 20, 2013
07/13

by
Radosław Adamczak; Rafał Latała; Alexander E. Litvak; Alain Pajor; Nicole Tomczak-Jaegermann

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We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of expectation of the supremum of "symmetric exponential" processes compared to the Gaussian ones in the Chevet inequality. This is used to give sharp upper estimate for a quantity $\Gamma_{k,m}$ that controls uniformly the Euclidean operator norm of the sub-matrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random...

Source: http://arxiv.org/abs/1107.4066v1

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Jul 20, 2013
07/13

by
Radosław Adamczak; Rafał Latała; Alexander E. Litvak; Alain Pajor; Nicole Tomczak-Jaegermann

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We establish new tail estimates for order statistics and for the Euclidean norms of projections of an isotropic log-concave random vector. More generally, we prove tail estimates for the norms of projections of sums of independent log-concave random vectors, and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub-matrices in the setting of a log-concave ensemble. This is used to study a quantity $A_{k,m}$ that controls uniformly the operator norm...

Source: http://arxiv.org/abs/1107.4070v1

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Sep 22, 2013
09/13

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Radosław Adamczak; Rafał Latała; Alexander E. Litvak; Alain Pajor; Nicole Tomczak-Jaegermann

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We study the Restricted Isometry Property of a random matrix $\Gamma$ with independent isotropic log-concave rows. To this end, we introduce a parameter $\Gamma_{k,m}$ that controls uniformly the operator norm of sub-matrices with $k$ rows and $m$ columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of projections of an isotropic log-concave vector.

Source: http://arxiv.org/abs/1103.0401v1

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71

Sep 20, 2013
09/13

by
Radosław Adamczak; Rafał Latała; Alexander E. Litvak; Krzysztof Oleszkiewicz; Alain Pajor; Nicole Tomczak-Jaegermann

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We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\R^n$, stating that for every $t\geq 1$, $P(|X|\geq ct\sqrt n)\leq \exp(-t\sqrt n)$. More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(E|X|^p)^{1/p}\sim E |X|+\sup_{z\in S^{n-1}}(E | < z,X>|^p)^{1/p}$.

Source: http://arxiv.org/abs/1205.2515v1

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Sep 20, 2013
09/13

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Radosław Adamczak; Olivier Guédon; Rafał Latała; Alexander E. Litvak; Krzysztof Oleszkiewicz; Alain Pajor; Nicole Tomczak-Jaegermann

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Let $p\geq 1$, $\eps >0$, $r\geq (1+\eps) p$, and $X$ be a $(-1/r)$-concave random vector in $\R^n$ with Euclidean norm $|X|$. We prove that $(\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E|X|+\sigma_{p}(X))$, where $\sigma_{p}(X)=\sup_{|z|\leq 1}(\E||^{p})^{1/p}$, $C(\eps)$ depends only on $\eps$ and $c$ is a universal constant. Moreover, if in addition $X$ is centered then $(\E |X|^{-p})^{-1/{p}}\geq c(\eps) (\E|X| - C \sigma_{p}(X))$.

Source: http://arxiv.org/abs/1207.6618v1