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Brownian motion with a parabolic drift and airy functions
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  • Published: 01 February 1989

Brownian motion with a parabolic drift and airy functions

  • Piet Groeneboom1 

Probability Theory and Related Fields volume 81, pages 79–109 (1989)Cite this article

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Summary

Let {W(t): t ∈ ∝} be two-sided Brownian motion, originating from zero, and let V(a) be defined by V(a)=sup}t ∈ ∝: W(t)−(t−a)2 is maximal}. Then {V(a): a ∈ ℝ} is a Markovian jump process, running through the locations of maxima of two-sided Brownian motion with respect to the parabolas f a(t)=(t−a)2. We give an analytic expression for the infinitesimal generators of the processes a ∈ ℝ, in terms of Airy functions in Theorem 4.1. This makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators (i.e. estimators derived under order restrictions). An example of this is given in Groeneboom (1985), where the asymptotic distribution of the (standardized) L 1-distance between a decreasing density and the Grenander maximum likelihood estimator of this density is determined. On our way to Theorem 4.1 we derive some other results. For example, we give an analytic expression for the joint density of the maximum and the location of the maximum of the process {W(t)−ct 2: t ∈ ℝ}, where c is an aribrary positive constant. We also determine the Laplace transform of the integral over a Brownian excursion. These last results also have recently been derived by several other authors, using a variety of methods.

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Authors and Affiliations

  1. Department of Mathematics, Delft University of Technology, P.O. Box 356, 2600 AJ, Delft, The Netherlands

    Piet Groeneboom

Authors
  1. Piet Groeneboom
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Additional information

This paper was awarded the Rollo Davidson prize 1985 (Cambridge, UK)

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Groeneboom, P. Brownian motion with a parabolic drift and airy functions. Probab. Th. Rel. Fields 81, 79–109 (1989). https://doi.org/10.1007/BF00343738

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  • Received: 19 October 1984

  • Revised: 04 June 1988

  • Published: 01 February 1989

  • Issue Date: February 1989

  • DOI: https://doi.org/10.1007/BF00343738

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Keywords

  • Brownian Motion
  • Transition Density
  • Joint Density
  • Airy Function
  • Brownian Bridge
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