A Note on Weighted Approximations to the Uniform Empirical and Quantile Processes

Abstract

Recently, M. Csörgő, S. Csörgő, Horváth and Mason (1986a) obtained a weighted approximation to the uniform empirical and quantile processes by a sequence of Brownian bridges. The purpose of this note is to give a short and elementary proof of their weighted approximation to the uniform quantile process. Their corresponding weighted approximation to the uniform empirical process follows in a direct fashion from that of the imiform quantile process. The present proof, as was the former, is based on the Komlós, Major and Tusnády (1976) strong approximation to the partial sum process. It is shown, however, that almost the same weighted approximation to the uniform empirical and quantile processes can be derived from the older Skorokhod (1965) embedding. This alternate weighted approximation, obtained via the Skorokhod embedding, is likely to be sufficient for nearly all applications of the weighted approximation methodology and has the advantage that its proof is more suitable for instructional purposes.KeywordsAsymptotic DistributionWiener ProcessWeighted ApproximationStrong ApproximationBrownian BridgeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.