Small deviation estimates for norms of Gaussian vectors

Analysis Seminar
Wednesday, November 13, 2019 - 1:55pm for 1 hour (actually 50 minutes)
Skiles 005
Konstantin Tikhomirov – Georgia Tech
Shahaf Nitzan
Let |.| be a norm in R^n, and let G be the standard Gaussian vector.
We are interested in estimating from above the probabilities
P{|G|<(1-t)E|G|} in terms of t. For 1-unconditional norms
in the L-position, we prove small deviation estimates which match those for the
ell-infinity norm: in a sense, among all 1-unconditional norms in the L-position,
the left tail of |G| is the heaviest for ell-infinity. Results for general norms are also obtained.
The proof is based on an application of the hypercontractivity property combined with
certain transformations of the original norm.
Joint work with G.Paouris and P.Valettas.