### 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 double-struck E(K(s_{1}, t_{1}) K(s_{2}, t_{2})) = (s _{1} ∧ s_{2}-s_{1}s_{2}) t_{1} ∧ t_{2}, 0 ≤ s_{1}, s_{2} ≤ 1, t_{1}, t_{2} ≥ 0. For each t > 0, the process K(·, t) is a Brownian bridge on the scale of √t. Let M_{1}*(t) ≥ M_{2}*(t) ≥ ⋯ ≥ M_{i}*(t) ≥ ⋯ ≥ 0 be the ranked excursion heights of K(·, t). In this paper, we study the path properties of the process t → M_{i}*(t). Two laws of the iterated logarithm are established to describe the asymptotic behaviors of M_{i}* (t) as t goes to infinity.

Original language | English |
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Pages (from-to) | 145-163 |

Number of pages | 19 |

Journal | Journal of Theoretical Probability |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2004 |

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### Keywords

- Excursions
- Kiefer process
- Ranked heights

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Theoretical Probability*,

*17*(1), 145-163. https://doi.org/10.1023/B:JOTP.0000020479.46788.c9