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The limiting behaviour of the maximal spacing generated by an i.i.d. sequence of Gaussian random variables

Published online by Cambridge University Press:  14 July 2016

Paul Deheuvels
Paul Deheuvels*
Affiliation:
Université Paris VI
*
∗ Postal address: 7 Avenue du Château, 92340 Bourg-la-Reine, France.
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Abstract

Let Mn be the maximal spacing generated in the sample's range by Χ1,· ··, Χ n, independent and identically distributed Gaussian N(0, 1) random variables. We obtain the limiting distribution of and show that

according to whether ε> 0 or ε < 0.

Keywords

ORDER STATISTICSSTRONG LAWS. LAWS OF THE ITERATED LOGARITHMALMOST SURE CONVERGENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 4 , December 1985 , pp. 816 - 827
Copyright
Copyright © Applied Probability Trust 1985 

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References

Barndorff-Nielsen, O. (1961) On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables. Math. Scand. 9, 383–384.CrossRefGoogle Scholar
David, H. A. (1981) Order Statistics , 2nd edn. Wiley, New York.Google Scholar
Deheuvels, P. (1982a) Strong limiting bounds for maximal uniform spacings. Ann Prob. 10, 1058–1065.CrossRefGoogle Scholar
Deheuvels, P. (1982b) On record times associated with kth extremes, Proc. 3rd Pannonian Symp. Math. Statist. , ed. Mogyorodi, J., Vincze, I. and Wertz, W., Akademiai Kiado, Budapest, 43–51.Google Scholar
Deheuvels, P. (1983a) The complete characterization of the upper and lower class of the record and inter-record times of an i.i.d. sequence. Z. Wahrscheinlichkeitsth. 62, 1–6.Google Scholar
Deheuvels, P. (1983b) The strong approximation of extremal processes (II). Z. Wahrscheinlichkeitsth. 62, 7–15.Google Scholar
Devroye, L. (1981) Laws of the iterated logarithm for order statistics of uniform spacings. Ann. Prob. 9, 860–867.Google Scholar
Devroye, L. (1983) The largest exponential spacing. Utilitas Math. Google Scholar
Henrici, P. (1977) Applied and Computational Complex Analysis. Wiley, New York.Google Scholar
Kiefer, J. (1973) Iterated logarithm analogues for sample quantiles when pn ? 0. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 227–244.Google Scholar
Malmquist, S. (1950) On a property of order statistics from a rectangular distribution. Skand. Aktuarietidskr. 33, 214–222.Google Scholar
Slud, E. (1978) Entropy and maximal spacings for random partitions. Z. Wahrscheinlichkeitsth. 41, 341–352.CrossRefGoogle Scholar
Strawderman, W. E. and Holmes, P. T. (1970) On the law of the iterated logarithm for inter-record times. J. Appl. Prob. 7, 432–439.CrossRefGoogle Scholar
Sukhatme, P. V. (1937) Tests of significance for samples of the ?2 population with two degrees of freedom. Ann. Eugenics 8, 52–56.CrossRefGoogle Scholar
Wertz, W. and Schneider, B. (1979) Statistical density estimation: A bibliography. Internat. Statist. Rev. 47, 155–175.Google Scholar