Strong Limit Theorems for Weighted Quantile Processes

Abstract

A thorough description of the almost sure behavior of weighted uniform quantile processes is given. This includes analogues of nearly all known results for weighted uniform empirical processes, such as the James functional law of the iterated logarithm and the Csaki results on the supremum of the standardized empirical process. Subject to the usual regularity conditions, our results extend to the nonuniform quantile process. Also, in the process of obtaining our results, we derive an extension of a theorem of Kiefer, which is likely to be of independent interest.