Weak convergence of quantile and expectile processes under general assumptions

We show weak convergence of quantile and expectile processes to Gaussian limit processes in the space of bounded functions endowed with an appropriate semimetric which is based on the concepts of epi-and hypo- convergence as introduced in A. Bucher, J. Segers and S. Volgushev (2014), 'When Uniform Weak Convergence Fails: Empirical Processes for Dependence Functions and Residuals via Epi- and Hypographs' , Annals of Statistics 42. We impose assumptions for which it is known that weak convergence with respect to the supremum norm generally fails to hold. For quantiles, we consider stationary observations, where the marginal distribution function is assumed to be strictly increasing and continuous except for finitely many points and to admit strictly positive - possibly infinite - left- and right-sided derivatives. For expectiles, we focus on independent and identically distributed (i.i.d.) observations. Only a finite second moment and continuity at the boundary points but no further smoothness properties of the distribution function are required. We also show consistency of the bootstrap for this mode of convergence in the i.i.d. case for quantiles and expectiles.