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Scanning Brownian Processes

Part of:Limit theoremsStochastic 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 + A 0 of some ‘scanning set' A 0. 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 A 0. We ask if the set A 0 determines the scanning process, and investigate when, and how, it is possible to recover the structure of A 0 from realisations of the sample paths of the random field Z.

Keywords

SCAN PROCESSESBROWNIAN SHEETQUADRATIC VARIATIONGAUSSIAN PROCESSESCONVEX POLYGONS

MSC classification

Primary:60G15: Gaussian processes60G17: Sample path properties
Secondary:60F15: Strong theorems60G60: Random fields

Information

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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