On the general Bahadur-Kiefer quantile, and Vervaat processes:: Old and new

Adopting E. Parzen's ideas of 1979, we assume that the density-quantile function f o F-1 is continuous on (0, 1) and regularly varying in neighbourhoods of 0 and 1. Under this assumption we then prove strong and weak limit theorems for the general Bahadur-Kiefer and quantile processes. In particular, our investigations provide a new look at results obtained by G. R. Shorack in 1972, as well as throw a new light on those previously proved under a condition introduced by M. Csorgo and P. Revesz in 1978. The problem of constructing asymptotic confidence bands for the general quantile function F-1 is also discussed in detail, and in a historical context as well. Furthermore, the above mentioned results concerning the general Bahadur-Kiefer and quantile processes play a decisive role when investigating the asymptotic behaviour of the general Vervaat process V-n. The herein obtained strong and weak convergence results for the process V-n supplement and generalize the only known results so far in the area that were obtained by W. Vervaat in 1972.