On the ranked excursion heights of a Kiefer process

Endre Csáki, Yueyun Hu

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1 Citation (Scopus)

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(s1, t1) K(s2, t2)) = (s 1 ∧ 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 √t. Let M1*(t) ≥ M2*(t) ≥ ⋯ ≥ Mi*(t) ≥ ⋯ ≥ 0 be the ranked excursion heights of K(·, t). In this paper, we study the path properties of the process t → Mi*(t). Two laws of the iterated logarithm are established to describe the asymptotic behaviors of Mi* (t) as t goes to infinity.

Original languageEnglish
Pages (from-to)145-163
Number of pages19
JournalJournal of Theoretical Probability
Volume17
Issue number1
DOIs
Publication statusPublished - 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

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