Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-17T17:18:55.193Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Proof of Some Results of Rényi and the Asymptotic Distribution of the Range of his Kolmogorov-Smirnov Type Random Variables

Published online by Cambridge University Press:  20 November 2018

Miklós Csörgö
Miklós Csörgö*
Affiliation:
McGill University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X1 X2, … , Xn be mutually independent random variables with a common continuous distribution function F(t). Let Fn(t) be the corresponding empirical distribution function, that is Fn(t) = (number of Xi ⩽ t, 1 ⩽ i ⩽ n)/n.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 550 - 558
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Baxter, G. and Donsker, M. D., On the distribution of the supremum functional for processes with stationary independent increments, Trans. Amer. Math. Soc., 85 (1957), 73–87.Google Scholar
2. Csörgö, M., Some Rényi type limit theorems for empirical distribution functions, Ann. Math. Statist., 36 (1965), 322–326.Google Scholar
3. Csörgö, M., Some Smirnov-type theorems of probability theory, Ann. Math. Statist., 36 (1965), 1113–1119.Google Scholar
4. Csörgö, M., K-sample analogues of Rênyi's Kolmogorov-Smirnov type theorems, Bull. Amer. Math. Soc., 71 (1965), 616–618.Google Scholar
5. Csörgö, M., Some k-sample Kolmogorov-Smirnov-Rényi type theorems for empirical distribution functions, Acta Math. Acad. Sci. Hungar., 17, No. 3-4 (1966), 325–334.Google Scholar
6. Donsker, M. D., Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statist., 23 (1952), 277–281.Google Scholar
7. Doob, J. L., Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statist., 20 (1949), 393–403.Google Scholar
8. Feller, W., The asymptotic distribution of the range of independent random variables, Ann. Math. Statist., 22 (1951), 427–432.Google Scholar
9. Kac, M. and Pollard, H., The distribution of the maximum of partial sums of independent random variables, Can. J. Math., 2 (1950), 375–384.Google Scholar
10. Kuiper, N. H., Tests concerning random points on a circle, Proc. Nederl. Akad. Wetensch. Indag. Math., Ser. A, 63 (1960), 38–47.Google Scholar
11. Levy, P., Processus stochastiques et mouvement Brownien (Paris, 1948).Google Scholar
12. Rényi, A., On the theory of order statistics, Acta Math. Acad. Sci. Hungar., 4 (1953), 191–231.Google Scholar