Half-space depth of log-concave probability measures
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Published:2023-11-29
Issue:1-2
Volume:188
Page:309-336
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ISSN:0178-8051
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Container-title:Probability Theory and Related Fields
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language:en
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Short-container-title:Probab. Theory Relat. Fields
Author:
Brazitikos Silouanos,Giannopoulos Apostolos,Pafis Minas
Abstract
AbstractGiven a probability measure $$\mu $$
μ
on $${{\mathbb {R}}}^n$$
R
n
, Tukey’s half-space depth is defined for any $$x\in {{\mathbb {R}}}^n$$
x
∈
R
n
by $$\varphi _{\mu }(x)=\inf \{\mu (H):H\in {{{\mathcal {H}}}}(x)\}$$
φ
μ
(
x
)
=
inf
{
μ
(
H
)
:
H
∈
H
(
x
)
}
, where $$\mathcal{H}(x)$$
H
(
x
)
is the set of all half-spaces H of $${{\mathbb {R}}}^n$$
R
n
containing x. We show that if $$\mu $$
μ
is a non-degenerate log-concave probability measure on $${{\mathbb {R}}}^n$$
R
n
then $$\begin{aligned} e^{-c_1n}\leqslant \int _{{\mathbb {R}}^n}\varphi _{\mu }(x)\,d\mu (x) \leqslant e^{-c_2n/L_{\mu }^2} \end{aligned}$$
e
-
c
1
n
⩽
∫
R
n
φ
μ
(
x
)
d
μ
(
x
)
⩽
e
-
c
2
n
/
L
μ
2
where $$L_{\mu }$$
L
μ
is the isotropic constant of $$\mu $$
μ
and $$c_1,c_2>0$$
c
1
,
c
2
>
0
are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of $$L_q$$
L
q
-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.
Funder
Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni
University of Athens
Publisher
Springer Science and Business Media LLC
Subject
Statistics, Probability and Uncertainty,Statistics and Probability,Analysis
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