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We prove a new concentration inequality for U-statistics of order two foruniformly ergodic Markov chains. Working with bounded and $\pi$-canonicalkernels, we show that we can recover the convergence rate of Arcones andGin{\'e} who proved a concentration result for U-statistics of independentrandom variables and canonical kernels. Our result allows for a dependence ofthe kernels $h_{i,j}$ with the indexes in the sums, which prevents the use ofstandard blocking tools. Our proof relies on an inductive analysis where we usemartingale techniques, uniform ergodicity, Nummelin splitting and Bernstein'stype inequality. Assuming further that the Markov chain starts from itsinvariant distribution, we prove a Bernstein-type concentration inequality thatprovides sharper convergence rate for small variance terms.

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