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.
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.
We show that every symmetric random variable with log-concave tails satisfies the convex infimum convolution inequality with an optimal cost function (up to scaling). As a result, we obtain nearly optimal comparison of weak and strong moments for symmetric random vectors with independent coordinates with log-concave tails.
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