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1
Content available remote

Sudakov-type minoration for log-concave vectors

100%
Latała R.
Studia Mathematica
|
2014
|
tom 223
|
nr 3
251-274
EN
We formulate and discuss a conjecture concerning lower bounds for norms of log-concave vectors, which generalizes the classical Sudakov minoration principle for Gaussian vectors. We show that the conjecture holds for some special classes of log-concave measures and some weaker forms of it are satisfied in the general case. We also present some applications based on chaining techniques.
2
Content available remote

Tail and moment estimates for sums of independent random vectors with logarithmically concave tails

100%
Latała R.
Studia Mathematica
|
1996
|
tom 118
|
nr 3
301-304
EN
Let $X_i$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable $X = ∑v_{i}X_{i}$, where $v_i$ are vectors of some Banach space. We derive approximate formulas for the tail and moments of ∥X∥. The estimates are exact up to some universal constant and they extend results of S. J. Dilworth and S. J. Montgomery-Smith [1] for the Rademacher sequence and E. D. Gluskin and S. Kwapień [2] for real coefficients.
3
Content available remote

On Weak Tail Domination of Random Vectors

100%
Latała R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
tom 57
|
nr 1
75-80
EN
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 then weak tail domination implies strong tail domination. In particular, a positive answer to Oleszkiewicz's question would follow from the so-called Bernoulli conjecture. We also prove that any unconditional logarithmically concave distribution is strongly dominated by a product symmetric exponential measure.
4
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A note on the Ehrhard inequality

100%
Latała R.
Studia Mathematica
|
1996
|
tom 118
|
nr 2
169-174
EN
We prove that for λ ∈ [0,1] and A, B two Borel sets in $ℝ^n$ with A convex, $Φ^{-1}(γ_n(λA + (1-λ)B)) ≥ λΦ^{-1}(γ_n(A)) + (1-λ)Φ^{-1}(γ_n(B))$, where $γ_n$ is the canonical gaussian measure in $ℝ^n$ and $Φ^{-1}$ is the inverse of the gaussian distribution function.
5
Content available remote

Tail and moment estimates for some types of chaos

100%
Latała R.
Studia Mathematica
|
1999
|
tom 135
|
nr 1
39-53
EN
Let $X_i$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable $X= ∑_{i ≠ j}a_{i,j}X_iX_j$, where $a_{i,j}$ are real numbers. We derive approximate formulas for the tails and moments of X and of its decoupled version, which are exact up to some universal constants.
6
Content available remote

Weak and strong moments of random vectors

100%
Latała R.
Banach Center Publications
|
2011
|
tom 95
|
nr 1
115-121
EN
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.
7
Content available remote

Small ball probability estimates in terms of width

64%
Latała R., Oleszkiewicz K.
Studia Mathematica
|
2005
|
tom 169
|
nr 3
305-314
EN
A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body K ⊆ ℝⁿ with inradius w and γₙ(K) ≤ 1/2 we have $γₙ(sK) ≤ (2s)^{w²/4}γₙ(K)$ for any s ∈ [0,1], where γₙ is the standard Gaussian probability measure. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.
8
Content available remote

A note on sums of independent uniformly distributed random variables

64%
Latała R., Oleszkiewicz K.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 2
197-206
9
Content available remote

On the best constant in the Khinchin-Kahane inequality

64%
Latała R., Oleszkiewicz K.
Studia Mathematica
|
1994
|
tom 109
|
nr 1
101-104
EN
We prove that if $r_i$ is the Rademacher system of functions then $(ʃ ∥∑_{i=1}^{n} x_{i}r_{i}(t)∥^2 dt)^{1/2} ≤ √2 ʃ ∥∑_{i=1}^{n}x_{i}r_{i}(t)∥dt$ for any sequence of vectors $x_i$ in any normed linear space F.
10
Content available remote

Chevet type inequality and norms of submatrices

45%
Adamczak R., Latała R., Litvak A., Pajor A., Tomczak-Jaegermann N.
Studia Mathematica
|
2012
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tom 210
|
nr 1
35-56
EN
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 the expectation of the supremum of "symmetric exponential" processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity $Γ_{k,m}$ that controls uniformly the Euclidean operator norm of the submatrices with k rows and m columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the restricted isometry constant of a random matrix with independent log-concave unconditional rows. We also show that our Chevet type inequality does not extend to general isotropic log-concave random matrices.
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