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Let $X_i$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable $X=\sum v_i{X}_i$, 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.

We prove that for λ ∈ [0,1] and A, B two Borel sets in ${ℝ}^n$ with A convex, ${\Phi }^{-1}\left({\gamma }_n(\lambda A+(1-\lambda )B)\right)\ge \lambda {\Phi }^{-1}\left({\gamma }_n\left(A\right)\right)+(1-\lambda ){\Phi }^{-1}\left({\gamma }_n\left(B\right)\right)$, where ${\gamma }_n$ is the canonical gaussian measure in ${ℝ}^n$ and ${\Phi }^{-1}$ is the inverse of the gaussian distribution function.

Let $X_i$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable $X={\sum }_{i\ne j}{a}_{i,j}{X}_i{X}_j$, 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.

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.

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.

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.

We prove that if $r_i$ is the Rademacher system of functions then ${\left(ʃ\parallel {\sum }_{i=1}^n{x}_i{r}_i\left(t\right){\parallel }^{2}dt\right)}^{1/2}\le \surd 2ʃ\parallel {\sum }_{i=1}^n{x}_i{r}_i\left(t\right)\parallel dt$ for any sequence of vectors $x_i$ in any normed linear space F.

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 $\gamma ₙ\left(sK\right)\le {\left(2s\right)}^{w²/4}\gamma ₙ\left(K\right)$ 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.

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.

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 ${\Gamma }_{k,m}$ 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...