ψα-ESTIMATES FOR MARGINALS OF LOG-CONCAVE PROBABILITY MEASURES

We show that a random marginal πF (μ) of an isotropie log-concave probability measure μ on ℝ n exhibits better ψ α -behavior. For a natural variant ψ' α of the standard ψ α -norm we show the following: (i) If $k \le \sqrt n $ , then for a random F ∊ G n,k we have that πF(μ) is a ψ'₂-measure. We complement this result by showing that a random πF(μ) is, at the same time, super-Gaussian. (ii) If k = n δ , ½ < δ < 1, then for a random F ∊ G n,k we have that πF(μ) is a ψ' α(δ) -measure, where $\[\alpha \left( \delta \right) = \frac{{2\delta }}{{3\delta - 1}}]$ .