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On the Ranked Excursion Heights of a Kiefer Process

Author

Listed:
  • Endre Csáki

    (Hungarian Academy of Sciences)

  • Yueyun Hu

    (Université)

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.

Suggested Citation

  • Endre Csáki & Yueyun Hu, 2004. "On the Ranked Excursion Heights of a Kiefer Process," Journal of Theoretical Probability, Springer, vol. 17(1), pages 145-163, January.
    • Handle: RePEc:spr:jotpro:v:17:y:2004:i:1:d:10.1023_b:jotp.0000020479.46788.c9
    DOI: 10.1023/B:JOTP.0000020479.46788.c9
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    References listed on IDEAS

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    1. Endre Csáki & Yueyun Hu, 2001. "Asymptotic Properties of Ranked Heights in Brownian Excursions," Journal of Theoretical Probability, Springer, vol. 14(1), pages 77-96, January.
    2. Csáki, Endre & Shi, Zhan, 1998. "Some liminf results for two-parameter processes," Stochastic Processes and their Applications, Elsevier, vol. 78(1), pages 27-46, October.
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