Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains - Archive ouverte HAL
Journal Articles Bernoulli Year : 2022

Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains

Abstract

We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $\pi$-canonical kernels, we show that we can recover the convergence rate of Arcones and Giné who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein's type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.
Fichier principal
Vignette du fichier
paper.pdf (413.84 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03014763 , version 1 (20-11-2020)
hal-03014763 , version 2 (17-02-2021)
hal-03014763 , version 3 (28-05-2021)
hal-03014763 , version 4 (16-03-2022)

Identifiers

Cite

Quentin Duchemin, Yohann de Castro, Claire Lacour. Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Bernoulli, 2022, 29 (2), pp.929-956. ⟨10.3150/22-BEJ1485⟩. ⟨hal-03014763v4⟩
554 View
381 Download

Altmetric

Share

More