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

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 Gin{e} 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 Bernsteins 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.