Weak and strong moments of ℓᵣ-norms of log-concave vectors

Abstract

Weak and strong moments of ℓ r -norms of log-concave vectors * Rafa l Lata la and Marta Strzelecka revised version Abstract We show that for p ≥ 1 and r ≥ 1 the p-th moment of the ℓ r -norm of a logconcave 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.

Comparison of Weak and Strong Moments for Vectors With Independent Coordinates

“…Paouris [12] showed that (1.2) holds for log-concave vectors and sets T being balls in Euclidean spaces (see also [1]). This was generalized in [8] to balls in L r -spaces with 1 ≤ r < ∞. Unfortunately there are very few classes of log-concave vectors such that (1.2) is known to be satisfied for all sets T -this includes vectors uniformly distributed on l n r -balls (with 1 ≤ r ≤ ∞).…”

Estimates of norms of log-concave random matrices with dependent entries

“…In this article we will see how to generalise the main result of [6] to a wide class of random matrices with independent uncorrelated log-concave rows, following the scheme of proof of the original theorem from [6]. In order to obtain the key estimates for log-concave vectors needed in the proof we use the comparison of weak and strong moments of ℓ p -norm of X from [11] and a Sudakov minoration-type bound from [10].…”

Hadamard products and moments of random vectors

“…While the one-sided estimate σ p (X) ≤ M p (X) follows trivially from the fact that x = sup t * ≤1 | t, x |, obtaining the reverse bounds turns out to be much more challenging. As an example let us mention the Paouris inequality M p (X) ≤ C(M 1 (X) + σ p (X)) valid for the standard Euclidean norm and arbitrary log-concave random vector X in R n , see [18] and [1] (see also [14] for an extension of this result to a larger class of norms). Here and in the sequel C denotes an absolute constant, whose value may change at each occurrence.…”