THE BAHADUR-KIEFER REPRESENTATION OF THE TWO DIMENSIONAL SPATIAL MEDIANS

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.