Table Of ContentTitle Pages
New Perspectives in Stochastic Geometry
Wilfrid S. Kendall and Ilya Molchanov
Print publication date: 2009
Print ISBN-13: 9780199232574
Published to Oxford Scholarship Online: February 2010
DOI: 10.1093/acprof:oso/9780199232574.001.0001
Title Pages
(p.i) New Perspectives in Stochastic Geometry (p.ii)
(p.iii) New Perspectives in Stochastic Geometry
(p.iv)
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Dedication
New Perspectives in Stochastic Geometry
Wilfrid S. Kendall and Ilya Molchanov
Print publication date: 2009
Print ISBN-13: 9780199232574
Published to Oxford Scholarship Online: February 2010
DOI: 10.1093/acprof:oso/9780199232574.001.0001
Dedication
(p.v) In memory of David Kendall, 15 January 1918–23 October 2007 (p.vi)
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Preface
New Perspectives in Stochastic Geometry
Wilfrid S. Kendall and Ilya Molchanov
Print publication date: 2009
Print ISBN-13: 9780199232574
Published to Oxford Scholarship Online: February 2010
DOI: 10.1093/acprof:oso/9780199232574.001.0001
(p.vii) Preface
The roots of stochastic geometry derive from classical highlights of geometric
probability. Examples include concepts arising from explorations of the notion of
‘natural' probability, questions raised in nineteenth‐century UK recreational
mathematics, and the famous problem of the eighteenth‐century French
polymath Georges‐Louis Leclerc, Comte de Buffon. The actual phrase ‘stochastic
geometry' appears first to have been employed in its current sense by D.G.
Kendall and K. Krickeberg in 1969, arising from their planning for an
Oberwolfach workshop (Stoyan, Kendall and Mecke, 1987, foreword), though
Andrew Wade has pointed out its use by Frisch and Hammersley (1963, p. 895)
as one of two possible terms to describe the study of ‘random irregular
structures' motivated by percolation problems. It is now plain that stochastic
geometry has woven together various strands from a wide‐ranging variety of
sources, of which we mention just four representative examples:
• impetus arising from the creation and development of geostatistics
and mathematical morphology by G. Matheron and J. Serra and other
French workers;
• the German school of point process and queueing theory, involving
such names as J. Kerstan, K. Matthes, J. Mecke and K. Krickeberg;
• British and Australian research, growing out of the study of
classical geometric probabilities, and developing relationships with
stochastic analysis;
• the Armenian school of combinatorial geometry led by R.V.
Ambartzumian.
After 40 years stochastic geometry is now an established part of probability
theory, expounded in several well‐established monographs, and finding vital
employment in numerous application areas such as spatial statistics, image
analysis, materials science, and even finance. In common with many areas of
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Preface
applied mathematics, the subject has been heavily influenced by the availability
of cheap and powerful computing, justifying deeper study of models which are
flexible but computationally demanding, and providing opportunities for
simulation study of systems which would otherwise be inaccessible. A subtle and
expressive mathematical vocabulary has been developed, allowing us to describe
and investigate random patterns of points, lines, planes, fibres, surfaces,
tessellations and sets. A wealth of hard theoretical problems can be found in the
wide variety of application areas, and it is an ongoing challenge for stochastic
geometry to contribute effectively to the intense and powerful scientific effort
devoted to dealing with these problems.
The present volume has been motivated by the sense that stochastic geometry is
now poised for a further phase of development, acquiring new perspectives
(p.viii) from revival of its classical roots, from new connections with recent
developments in the main body of probability, and stimulated by the plethora of
rapidly developing applications. The purpose of the volume is therefore to
present recent developments so as to form an entry point into stochastic
geometry for mathematical scientists from other areas who are curious about
what it might offer, and to provide a resource for young researchers who wish to
engage with the subject and seek out areas in which they might make an original
contribution. The editors gathered together a group of experts, each of whom
was tasked with writing a chapter to cover a particular aspect of stochastic
geometry, reflecting the current state of the art but also commenting on and
providing new developments. To this end, the volume is divided into a chapter on
the fundamentals, and then into four parts. The initial Chapter 1 surveys what
can be described as classical stochastic geometry, summarizing its historical
development and clarifying the core mathematical concepts which arise
throughout stochastic geometry as a whole. Classical stochastic geometry has
itself recently seen a revival; Part I (‘New Developments in Classical Stochastic
Geometry’) contains four chapters describing new developments in random
polytopes, random measures, limit theory and tessellations (Chapters 2, 3, 4, 5).
In addition, recently there has arisen a number of strong connections with other
areas of mathematics such as percolation, random network theory and fractals,
forming the three Chapters 6, 7 and 8 of Part II ‘Stochastic Geometry and
Modern Probability'. There has always been a strong statistical theme in
stochastic geometry, represented here by Part III ‘Statistics and Stochastic
Geometry'. This commences with Chapter 9, discussing inferential issues arising
when one seeks to estimate parameters of random pattern models, and
continues with three further Chapters 10, 11 and 12, respectively covering
statistical shape, estimation of sets, and notions of data depth. Finally Part IV
‘Applications’ contains five chapters surveying applications in image analysis
(Chapter 13), stereology (Chapter 14), materials science (Chapter 15),
telecommunications (Chapter 16) and finance (Chapter 17); the growth of
stochastic geometry has always been fuelled by ever‐changing demands from
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Preface
application areas, and these chapters demonstrate strong possibilities for future
growth.
The authors and editors offer this volume to the mathematical community as a
stepping stone towards the new perspectives which stochastic geometry now
offers.
References in the Preface
Frisch, H. L. and Hammersley, J. M. (1963). Percolation processes and related
topics. J. Soc. Indust. Appl. Math., 11, 894–918.
Stoyan, D., Kendall, W. S., and Mecke, J. (1987). Stochastic Geometry and its
Applications. Wiley Series in Probability and Mathematical Statistics: Applied
Probability and Statistics. John Wiley & Sons Ltd., Chichester. With a foreword
by D. G. Kendall.
Page 3 of 3
Acknowledgements
New Perspectives in Stochastic Geometry
Wilfrid S. Kendall and Ilya Molchanov
Print publication date: 2009
Print ISBN-13: 9780199232574
Published to Oxford Scholarship Online: February 2010
DOI: 10.1093/acprof:oso/9780199232574.001.0001
(p.ix) Acknowledgements
The editors and authors would like to thank the Oxford University Press staff for
help with the production of this volume.
They are also grateful to Mathematisches Forschungsinstitut Oberwolfach for
the opportunity to organize a workshop under the same name, which greatly
aided the process of preparation of this volume, nearly 40 years after the
Oberwolfach workshop at which the phrase ‘Stochastic Geometry' first saw the
light of day. (p.x)
Page 1 of 1
List of Contributors
New Perspectives in Stochastic Geometry
Wilfrid S. Kendall and Ilya Molchanov
Print publication date: 2009
Print ISBN-13: 9780199232574
Published to Oxford Scholarship Online: February 2010
DOI: 10.1093/acprof:oso/9780199232574.001.0001
(p.xix) List of Contributors
Dr. Pierre Calka
(pierre.calka©math‐info.univ‐ paris5.fr)
MAP5, Université Paris
Descartes, 45,
rue des Saints‐Pères, 75270 Paris
Cedex 06, France.
Dr. Ignacio Cascos
(ignacio.cascos©uc3m.es)
Department of Statistics,
Universidad Carlos III de Madrid,
Av. Universidad 30, E‐28911
Leganés (Madrid), Spain.
Prof. Dr. Antonio Cuevas
(antonio.cuevas©uam.es)
Department of Mathematics,
Facultad de Ciencias,
Universidad Autonoma de Madrid,
28049 Madrid, Spain.
Prof. Dr. R. Fraiman
(rfraiman©cmat.edu.uy)
Centro de Matematica,
Universidad de la Republica,
Eduardo Acevedo, 1139,
Montevideo, Uruguay.
Prof. Dr. Remco van der Hofstad
(rhofstad©win.tue.nl)
Eindhoven University of Technology
Department of Mathematics and
Page 1 of 4
List of Contributors
Computer Science, HG 9.04
P.O. Box 513, 5600 MB Eindhoven,
the Netherlands.
Prof. Dr. W.S. Kendall
(w.s.kendall©warwick.ac.uk)
Dept of Statistics,
University of Warwick
Coventry CV4 7AL, United
Kingdom.
Prof. Dr. Günter Last
(last©math.uni‐karlsruhe.de)
Institut für Stochastik,
Universität Karlsruhe, D‐76128
Karlsruhe, Germany.
Dr. Huiling Le
(huiling.le©nottingham.ac.uk)
School of Mathematical Sciences,
University of Nottingham
Nottingham NG7 2RD, United
Kingdom.
Dr. M.N.M. van Lieshout
(colette©cwi.nl)
CWI, Science Park 123, NL‐1098,
XG Amsterdam, The Netherlands.
Prof. Dr. Klaus Mecke
(klaus.mecke©physik.uni‐ erlangen.de)
Institut für Theoretische Physik,
Universität Erlangen‐Nürnberg,
Staudtstrasse 7, 91058 Erlangen,
Germany.
Prof. Dr. Ilya Molchanov
(ilya©stat.unibe.ch)
IMSV, University of Bern
Sidlerstrasse 5, 3012 Bern,
Switzerland.
(p.xx)
Prof. Dr. Jesper Møller
(jm math.aau.dk)
Department of Mathematical
Sciences, Aalborg University,
F. Bajers Vej 7G,
DK‐9220 Aalborg Øst, Denmark.
Prof. Dr. Peter Mörters
(maspm©bath.ac.uk)
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