Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains
This work provides a theoretical improvement in concentration inequalities for U-statistics under dependency, which is important for researchers working with dependent data.
This paper establishes a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains, achieving the same convergence rate as results for independent random variables and canonical kernels. It also provides a sharper Bernstein-type concentration inequality for small variance terms when the Markov chain starts from its invariant distribution.
We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $π$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{é} 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.