Empirical Processes with a Bounded Ψ1 Diameter

Shahar Mendelson

doi:10.1007/s00039-010-0084-5

Empirical Processes with a Bounded Ψ₁ Diameter

Shahar Mendelson^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

39 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	988-1027
Number of pages	40
Journal	Geometric and Functional Analysis
Volume	20
Issue number	4
DOIs	https://doi.org/10.1007/s00039-010-0084-5
Publication status	Published - 2010

Access to Document

10.1007/s00039-010-0084-5

Cite this

@article{776910e1f530435290ee20931a9df089,

title = "Empirical Processes with a Bounded Ψ1 Diameter",

abstract = "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.",

keywords = "Empirical processes, generic chaining",

author = "Shahar Mendelson",

year = "2010",

doi = "10.1007/s00039-010-0084-5",

language = "English",

volume = "20",

pages = "988--1027",

journal = "Geometric and Functional Analysis",

issn = "1016-443X",

publisher = "Birkhauser",

number = "4",

}

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 - https://www.scopus.com/pages/publications/77958490141

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 -

Empirical Processes with a Bounded Ψ₁ Diameter

Abstract

Access to Document

Other files and links

Fingerprint

Cite this