On the Ranked Excursion Heights of a Kiefer Process

Endre Csáki () and Yueyun Hu ()
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Endre Csáki: Hungarian Academy of Sciences
Yueyun Hu: Université

Journal of Theoretical Probability, 2004, vol. 17, issue 1, 145-163

Abstract: Abstract Let (K(s,t), 0≤s≤1, t≥1) be a Kiefer process, i.e., a continuous two-parameter centered Gaussian process indexed by [0,1]×ℝ+ whose covariance function is given by $$\mathbb{E}$$ (K(s1,t1) K(s2,t2))=(s1∧s2-s1s2)t1∧t2, 0⩽s1, s2⩽1, t1, t2⩾ 0. For each t>0, the process K(·,t) is a Brownian bridge on the scale of $$\sqrt t$$ . Let M 1 * (t)⩾ M 2 * (t)⩾⋯⩾ M j * (t)⩾⋯⩾ 0 be the ranked excursion heights of K(ċ,t). In this paper, we study the path properties of the process t→M j * (t). Two laws of the iterated logarithm are established to describe the asymptotic behaviors of M j * (t) as t goes to infinity.

Keywords: Kiefer process; excursions; ranked heights (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1023/B:JOTP.0000020479.46788.c9

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