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THE BAHADUR-KIEFER REPRESENTATION OF

THE TWO DIMENSIONAL SPATIAL MEDIANS

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MIGUEL A. ARCONES

*Department of Mathematics, University of Texas, Austin, TX 78712-1082, U.S.A.*
(Received January 8, 1996; revised February 26, 1997)

**Abstract.**
We consider the asymptotic
behavior, both in distribution and almost sure, of the
Bahadur-Kiefer representation of the two dimensional
spatial medians. The rates appearing in this expansion
are non-standard. The rate in the almost sure expansion
is *n*(2 log *n*)^{-1/2}(2 log log *n*)^{-1}. The set of
clusters points in the almost sure representation is
obtained. The distribution of the Bahadur-Kiefer
representation of the two dimensional spatial medians
converges with rate *n*(2 log *n*)^{-1/2} to a limit that
is determined precisely.

*Key words and phrases*:
Bahadur-Kiefer
representations, spatial medians, empirical processes.

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