  We study the split Conformal Prediction method when applied to Markovian
data. We quantify the gap in terms of coverage induced by the correlations in
the data (compared to exchangeable data). This gap strongly depends on the
mixing properties of the underlying Markov chain, and we prove that it
typically scales as $\sqrt{t_\mathrm{mix}\ln(n)/n}$ (where $t_\mathrm{mix}$ is
the mixing time of the chain). We also derive upper bounds on the impact of the
correlations on the size of the prediction set. Finally we present $K$-split
CP, a method that consists in thinning the calibration dataset and that adapts
to the mixing properties of the chain. Its coverage gap is reduced to
$t_\mathrm{mix}/(n\ln(n))$ without really affecting the size of the prediction
set. We finally test our algorithms on synthetic and real-world datasets.
