The Annals of Statistics

Marginal Densities of the Least Concave Majorant of Brownian Motion

Chris Carolan and Richard Dykstra

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Abstract

A clean, closed form, joint density is derived for Brownian motion, its least concave majorant, and its derivative, all at the same fixed point. Some remarkable conditional and marginal distributions follow from this joint density. For example, it is shown that the height of the least concave majorant of Brownian motion at a fixed time point has the same distribution as the distance from the Brownian motion path to its least concave majorant at the same fixed time point. Also, it is shown that conditional on the height of the least concave majorant of Brownian motion at a fixed time point, the left-hand slope of the least concave majorant of Brownian motion at the same fixed time point is uniformly distributed.

Article information

Source
Ann. Statist., Volume 29, Number 6 (2001), 1732-1750.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015345960

Digital Object Identifier
doi:10.1214/aos/1015345960

Mathematical Reviews number (MathSciNet)
MR1891744

Zentralblatt MATH identifier
1044.60072

Subjects
Primary: 62E15: Exact distribution theory
Secondary: 62H10: Distribution of statistics

Keywords
Brownian motion least concave majorant stochastic ordering likelihood ratio ordering

Citation

Carolan, Chris; Dykstra, Richard. Marginal Densities of the Least Concave Majorant of Brownian Motion. Ann. Statist. 29 (2001), no. 6, 1732--1750. doi:10.1214/aos/1015345960. https://projecteuclid.org/euclid.aos/1015345960


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