TY - JOUR
T1 - Empirical Processes with a Bounded Ψ1 Diameter
AU - Mendelson, Shahar
PY - 2010
Y1 - 2010
N2 - We study the empirical process, where F is a class of mean-zero functions on a probability space (Ω, μ), and are selected independently according to μ. We present a sharp bound on this supremum that depends on the Ψ1 diameter of the class F (rather than on the Ψ2 one) and on the complexity parameter γ2(F,Ψ2). In addition, we present optimal bounds on the random diameters using the same parameters. As applications, we extend several well-known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on ℝn.
AB - We study the empirical process, where F is a class of mean-zero functions on a probability space (Ω, μ), and are selected independently according to μ. We present a sharp bound on this supremum that depends on the Ψ1 diameter of the class F (rather than on the Ψ2 one) and on the complexity parameter γ2(F,Ψ2). In addition, we present optimal bounds on the random diameters using the same parameters. As applications, we extend several well-known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on ℝn.
KW - Empirical processes
KW - generic chaining
UR - http://www.scopus.com/inward/record.url?scp=77958490141&partnerID=8YFLogxK
U2 - 10.1007/s00039-010-0084-5
DO - 10.1007/s00039-010-0084-5
M3 - Article
SN - 1016-443X
VL - 20
SP - 988
EP - 1027
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 4
ER -