I am trying to use kernels with ROC curve, and 'm succeed to plot them but now my query is about theoretical grounds, i.e. its bias, var, etc. I want to evaluate the theorems (1 & 2) in Pulit (2016). Shortly Proofs are available in Appendix, but I am facing issues to solving them. I want proofs of those theorems with detail and explanation. Kindly can any one can prove them with some detail. Regards.