Open Access
VOL. 1 | 2018 Chapter 22. Gaussian Approximations and Related Questions for the Spacings process
Gane Samb LO

Editor(s) Hamet SEYDI, Gane Samb LO, Aboubakary DIAKHABY


All the available results on the approximation of the k-spacings process to Gaussian processes have only used one approach, that is the Shorack and Pyke’s one. Here, it is shown that this approach cannot yield a rate better than $(N/ \log\log N)^{−\frac{1}{4}}(\log N)^{\frac{1}{2}}$. Strong and weak bounds for that rate are specified both where k is fixed and where $k \rightarrow + \infty$. A Glivenko-Cantelli Theorem is given while Stute’s result for the increments of the empirical process based on independent and identically distributed random variables is extended to the spacings process. One of the Mason- Wellner-Shorack cases is also obtained.


Published: 1 January 2018
First available in Project Euclid: 26 September 2019

Digital Object Identifier: 10.16929/sbs/2018.100-04-05

Primary: 60B10 , 60F15 , 60G30

Keywords: empirical process , gamma distribution and function , Law of the iterated logarithm , order statistics , oscillation modulus , spacings , strong and weak approximation

Rights: Copyright © 2018 The Statistics and Probability African Society

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