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A note on weighted approximations to the uniform empirical and quantile processes. (English) Zbl 0722.60042
Sums, Trimmed sums and extremes, Prog. Probab. 23, 269-283 (1991).
[For the entire collection see Zbl 0717.00017.]
M. Csörgö, S. Csörgö, L. Horváth and D. M. Mason [Ann. Probab. 14, 31-85 (1986; Zbl 0589.60029)] used a sequence of Brownian bridges to obtain a weighted approximation to uniform empirical and quantile processes. The author provides a short and elementary proof of their approximation to the uniform quantile process. Both his proof and the original proof are based on the KMT strong approximation to the partial sum process. However, the author shows that essentially the same approximation holds if Skorokhod embedding is used instead. The author emphasizes that this alternate formulation, while of equal value in applications, is likely to be more accessible to those not well versed in the KMT approximation.

MSC:
60G50 Sums of independent random variables; random walks
60F15 Strong limit theorems