Open Access
2023 On dimension-dependent concentration for convex Lipschitz functions in product spaces
Han Huang, Konstantin Tikhomirov
Author Affiliations +
Electron. J. Probab. 28: 1-23 (2023). DOI: 10.1214/23-EJP944


Let n1, K>0, and let X=(X1,X2,,Xn) be a random vector in Rn with independent K–subgaussian components. We show that for every 1–Lipschitz convex function f in Rn (the Lipschitzness with respect to the Euclidean metric), t>0,


where c>0 is a universal constant. The estimates are optimal in the sense that for every nC˜ and t>0 there exist a product probability distribution X in Rn with K–subgaussian components, and a 1–Lipschitz convex function f, with


The obtained deviation estimates for subgaussian variables are in sharp contrast with the case of variables with bounded Xiψp–quasi-norms for p(0,2).

Funding Statement

K.T. is partially supported by the Sloan Research Fellowship.


We would like to express our gratitude to the anonymous reviewers whose valuable feedback and suggestions greatly improved the quality of this paper.


Download Citation

Han Huang. Konstantin Tikhomirov. "On dimension-dependent concentration for convex Lipschitz functions in product spaces." Electron. J. Probab. 28 1 - 23, 2023.


Received: 24 June 2021; Accepted: 31 March 2023; Published: 2023
First available in Project Euclid: 3 May 2023

MathSciNet: MR4583676
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP944

Primary: 60F10

Keywords: concentration of measure , convex Lipschitz functions , subgaussian random variables

Vol.28 • 2023
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