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A functional equation technique for obtaining wiener process probabilities associated with theorems of Kolmogorov-Smirnov type

Published online by Cambridge University Press:  24 October 2008

J. Kiefer  and
D. V. Lindley
J. Kiefer
Affiliation:
Cornell and Oxford Universities
D. V. Lindley
Affiliation:
Cornell and Oxford Universities
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Extract

1. Introduction. Since the first proofs by Kolmogorov (13) and Smirnov ((14), (15)) of their well-known results on the limit distribution of the deviations of the sample distribution function, many alternative proofs of these results have been given. For example, we may cite the various approaches of Feller (4), Doob (3), Kac (8), Gnedenko and Korolyuk(7), and Anderson and Darling (1). The approaches of (3), (8) and (1) rest on a probabilistic computation regarding the Wiener process, and are justified by the paper of Donsker (2) (see also (11)). Of all these approaches, only those of (8) and (1) can be extended to obtain the limit distributions of the ‘k–sample’ generalizations of the Kolmogorov-Smirnov statistics suggested in (9), and the author ((9), (10)) and Gihman(6) carried out such proofs.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 55 , Issue 4 , October 1959 , pp. 328 - 332
Copyright
Copyright © Cambridge Philosophical Society 1959

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References

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