Comparison of Weak and Strong Moments for Vectors With Independent Coordinates

Abstract

We show that for p ≥ 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 ≥ 2. The latest condition turns out to be necessary in the i.i.d. case.

On the convex infimum convolution inequality with optimal cost function

“…Immediately we obtain the following corollary in the spirit of the results from [13,1,7,8]. Similar inequalities for Rademacher sums with the emphasis on exact values of constants have also been studied by Oleszkiewicz (see [12, Theorem 2.1]).…”

“…If we only assume that the coordinates of X are independent and their moments grow α-regularly, then (2.2) does not always hold (the example here is a vector with independent coordinates distributed like in (2.1); see Section 5 for details), although by [7, Theorem 1.1] it holds if we allow the constant at E X to be greater than 1 and to depend on α. Hence Corollary 2.5 and example (2.1) partially answer the following question raised in [7]: "For which vectors does the comparison of weak and strong moments hold with constant 1 at the first strong moment? "…”

“…Using the aforementioned novel tools from [3], we show that product measures with symmetric marginals having log-concave tails satisfy the optimal convex ICI, which complements Lata la and Wojtaszczyk's result about log-concave product measures. This has applications to concentration and moment comparison of any norm of such vectors in the spirit of celebrated Paouris' inequality (see [13] and [1]) and addresses some questions posed lately in [7]. We also offer an example showing that the assumption of log-concave tails cannot be weakened substantially.…”

Tail and moment estimates for a class of random chaoses of order two

“…Since any symmetric random variable X with log-concave tails satisfies (1) with α = 2, this generalizes the previous result of Latała [10]. Moreover (1) arises naturally in the paper of Latała and Strzelecka [9] as a sufficient condition (and even necessary in the i.i.d case) for comparison of weak and strong moments of the random variable sup t∈T ⊂R n t i X i . Lastly it is shown in [12] (see Remark A.11 below) that if ln P(|X| ≥ Ktx) ≤ t β P(|X| ≥ x) for any t, x ≥ 1 and some constants K, β, then (1) holds with α = α(K, β).…”

“…Although formulas are similar as in Latała's paper [10], we cannot use his approach since our random variables do not satisfy nice dimension-free concentration inequalities. Instead we use a recent result of Latała and Strzelecka [9] and reduce the question to finding a right bound on L 1 -norm of suprema. To treat this we use some ideas from [8] and [1].…”