  Training-conditional coverage guarantees in conformal prediction concern the
concentration of the error distribution, conditional on the training data,
below some nominal level. The conformal prediction methodology has recently
been generalized to the covariate shift setting, namely, the covariate
distribution changes between the training and test data. In this paper, we
study the training-conditional coverage properties of a range of conformal
prediction methods under covariate shift via a weighted version of the
Dvoretzky-Kiefer-Wolfowitz (DKW) inequality tailored for distribution change.
The result for the split conformal method is almost assumption-free, while the
results for the full conformal and jackknife+ methods rely on strong
assumptions including the uniform stability of the training algorithm.
