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Scanning Brownian Processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Robert J. Adler  and
Ron Pyke
Robert J. Adler*
Affiliation:
Technion—Israel Institute of Technology and University of North Carolina
Ron Pyke*
Affiliation:
University of Washington
*
∗ Postal address: Faculty of Industrial Engineering and Management, Technion—Israel Institute of Technology, Haifa, Israel 32000, and Department of Statistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3260, USA. e-mail: adler@stat.unc.edu, robert@ieadler.technion.ac.il
∗∗ Postal address: Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA. e-mail: pyke@math.washington.edu
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Abstract

The ‘scanning process' Z(t), t ∈ ℝk of the title is a Gaussian random field obtained by associating with Z(t) the value of a set-indexed Brownian motion on the translate t + A0 of some ‘scanning set' A0. We study the basic properties of the random field Z relating, for example, its continuity and other sample path properties to the geometrical properties of A0. We ask if the set A0 determines the scanning process, and investigate when, and how, it is possible to recover the structure of A0 from realisations of the sample paths of the random field Z.

Keywords

SCAN PROCESSESBROWNIAN SHEETQUADRATIC VARIATIONGAUSSIAN PROCESSESCONVEX POLYGONS

MSC classification

Primary: 60G15: Gaussian processes 60G17: Sample path properties
Secondary: 60F15: Strong theorems 60G60: Random fields
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 295 - 326
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Research supported in part by US–Israel Binational Science Foundation and Office of Naval Research.

Research supported in part by US–Israel Binational Science Foundation and NSF.

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