Functional limit laws for the increments of the quantile process; with applications

We establish a functional limit law of the logarithm for the increments of the normed quantile process based upon a random sample of size n → ∞. We extend a limit law obtained by Deheuvels and Mason (12), showing that their results hold uniformly over the bandwidth h, restricted to vary in [h n , h n ], where {h n } n≥1 and {h ′′ n } n≥1 are appropriate nonrandom sequences. We treat the case where the sample observations follow possibly non-uniform distributions. As a consequence of our theorems, we provide uniform limit laws for nearest-neighbor density estimators, in the spirit of those given by Deheuvels and Mason (13) for kernel-type estimators. AMS 2000 subject classifications: Primary 60F15, 60F17; secondary 62G07.