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ON THE LIMIT BEHAVIOUR OF THE JOINT

DISTRIBUTION FUNCTION OF ORDER STATISTICS

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H. FINNER AND M. ROTERS

*FB IV-Mathematik/Statistik, Universität Trier, D-54286 Trier, Germany*

(Received December 11, 1992; revised June 11, 1993)

**Abstract.**
For *k* \in _{0} fixed we consider the joint
distribution function *F*^{k}_{n} of the *n* - *k* smallest order statistics
of *n* real-valued independent, identically distributed random variables
with arbitrary cumulative distribution function *F*. The main result of
the paper is a complete characterization of the limit behaviour of
*F*^{k}_{n}(*x*_{1}, .... , *x*_{n-k}) in terms of the limit behaviour of *n*(1
- *F*(*x*_{n})) if *n* tends to infinity, i.e., in terms of the limit superior,
the limit inferior, and the limit if the latter exists. This
characterization can be reformulated equivalently in terms of the limit
behaviour of the cumulative distribution function of the (*k* + 1)-th
largest order statistic. All these results do not require any further
knowledge about the underlying distribution function *F*.

*Key words and phrases*:
Extreme order statistic, extreme
value distribution, joint distribution function of order statistics,
multiple comparisons, Poisson distribution, uniform distribution.

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