  Despite the ubiquity of U-statistics in modern Probability and Statistics,
their non-asymptotic analysis in a dependent framework may have been
overlooked. In a recent work, a new concentration inequality for U-statistics
of order two for uniformly ergodic Markov chains has been proved. In this
paper, we put this theoretical breakthrough into action by pushing further the
current state of knowledge in three different active fields of research. First,
we establish a new exponential inequality for the estimation of spectra of
trace class integral operators with MCMC methods. The novelty is that this
result holds for kernels with positive and negative eigenvalues, which is new
as far as we know. In addition, we investigate generalization performance of
online algorithms working with pairwise loss functions and Markov chain
samples. We provide an online-to-batch conversion result by showing how we can
extract a low risk hypothesis from the sequence of hypotheses generated by any
online learner. We finally give a non-asymptotic analysis of a goodness-of-fit
test on the density of the invariant measure of a Markov chain. We identify
some classes of alternatives over which our test based on the $L_2$ distance
has a prescribed power.
