Probability Theory and Related Fields

, Volume 81, Issue 1, pp 79-109

First online:

Brownian motion with a parabolic drift and airy functions

  • Piet GroeneboomAffiliated withDepartment of Mathematics, Delft University of Technology

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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)−(ta)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)=(ta)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.