Conditions for quantile process approximations

Csörgö and Révész introduced a condition on the density of a distribution function that is sufficient to obtain weighted approximations for the pertaining normalized quantile process. We prove that this condition implies the extended regular variation of the density quantile function and that therefore it is substantially stronger than another sufficient condition due to Shorack, which is implied by O-regular variation. The relationship between these conditions is clarified by introducing a new Csörgö–Révész type condition that is equivalent to O-regular variation. Then we show that the Csörgö–Révész condition is sufficient to establish stochastic and almost sure approximations of the tail quantile function, which were proven in previous papers under the stronger assumption that the density quantile function is regularly varying.