Service interruption on Monday 11 July from 12:30 to 13:00: all the sites of the CCSD (HAL, EpiSciences, SciencesConf, AureHAL) will be inaccessible (network hardware connection).

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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-03014763
Contributor : Quentin Duchemin Connect in order to contact the contributor
Submitted on : Wednesday, March 16, 2022 - 3:14:08 PM
Last modification on : Wednesday, April 13, 2022 - 9:33:27 AM

Files

paper.pdf
Files produced by the author(s)

Identifiers

• HAL Id : hal-03014763, version 4
• ARXIV : 2011.11435

Citation

Quentin Duchemin, yohann de Castro, Claire Lacour. Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, In press. ⟨hal-03014763v4⟩

Record views