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Stochastic geometry is the branch of mathematics that studies
geometric structures associated with random configurations, such as
random graphs, tilings and mosaics. Due to its close ties with
stereology and spatial statistics, the results in this area are
relevant for a large number of important applications, e.g. to the
mathematical modeling and statistical analysis of telecommunication
networks, geostatistics and image analysis. In recent years - due
mainly to the impetus of the authors and their collaborators - a
powerful connection has been established between stochastic
geometry and the Malliavin calculus of variations, which is a
collection of probabilistic techniques based on the properties of
infinite-dimensional differential operators. This has led in
particular to the discovery of a large number of new quantitative
limit theorems for high-dimensional geometric objects. This unique
book presents an organic collection of authoritative surveys
written by the principal actors in this rapidly evolving field,
offering a rigorous yet lively presentation of its many facets.
Stochastic geometry is the branch of mathematics that studies
geometric structures associated with random configurations, such as
random graphs, tilings and mosaics. Due to its close ties with
stereology and spatial statistics, the results in this area are
relevant for a large number of important applications, e.g. to the
mathematical modeling and statistical analysis of telecommunication
networks, geostatistics and image analysis. In recent years - due
mainly to the impetus of the authors and their collaborators - a
powerful connection has been established between stochastic
geometry and the Malliavin calculus of variations, which is a
collection of probabilistic techniques based on the properties of
infinite-dimensional differential operators. This has led in
particular to the discovery of a large number of new quantitative
limit theorems for high-dimensional geometric objects. This unique
book presents an organic collection of authoritative surveys
written by the principal actors in this rapidly evolving field,
offering a rigorous yet lively presentation of its many facets.
The concept of Wiener chaos generalizes to an infinite-dimensional
setting the properties of orthogonal polynomials associated with
probability distributions on the real line. It plays a crucial role
in modern probability theory, with applications ranging from
Malliavin calculus to stochastic differential equations and from
probabilistic approximations to mathematical finance. This book is
concerned with combinatorial structures arising from the study of
chaotic random variables related to infinitely divisible random
measures. The combinatorial structures involved are those of
partitions of finite sets, over which Moebius functions and related
inversion formulae are defined. This combinatorial standpoint
(which is originally due to Rota and Wallstrom) provides an ideal
framework for diagrams, which are graphical devices used to compute
moments and cumulants of random variables. Several applications are
described, in particular, recent limit theorems for chaotic random
variables. An Appendix presents a computer implementation in
MATHEMATICA for many of the formulae.
A self-contained and probability-oriented introduction to the
theory of lattice of partitions, with a unique software
implementation that makes our book an ideal introduction to the
field A complete and self-contained combinatorial analysis of
cumulants and diagram formulae, unique in its genre An introduction
to Wiener chaos, and a new combinatorial interpretation of recently
proved limit theorems The concept of Wiener chaos generalizes to an
infinite-dimensional setting the properties of orthogonal
polynomials associated with probability distributions on the real
line. It plays a crucial role in modern probability theory, with
applications ranging from Malliavin calculus to stochastic
differential equations and from probabilistic approximations to
mathematical finance. This book is concerned with combinatorial
structures arising from the study of chaotic random variables
related to infinitely divisible random measures. The combinatorial
structures involved are those of partitions of finite sets, over
which M bius functions and related inversion formulae are defined.
This combinatorial standpoint (which is originally due to Rota and
Wallstrom) provides an ideal framework for diagrams, which are
graphical devices used to compute moments and cumulants of random
variables. Several applications are described, in particular,
recent limit theorems for chaotic random variables. An Appendix
presents a computer implementation in MATHEMATICA for many of the
formulae
Stein's method is a collection of probabilistic techniques that
allow one to assess the distance between two probability
distributions by means of differential operators. In 2007, the
authors discovered that one can combine Stein's method with the
powerful Malliavin calculus of variations, in order to deduce
quantitative central limit theorems involving functionals of
general Gaussian fields. This book provides an ideal introduction
both to Stein's method and Malliavin calculus, from the standpoint
of normal approximations on a Gaussian space. Many recent
developments and applications are studied in detail, for instance:
fourth moment theorems on the Wiener chaos, density estimates,
Breuer-Major theorems for fractional processes, recursive cumulant
computations, optimal rates and universality results for
homogeneous sums. Largely self-contained, the book is perfect for
self-study. It will appeal to researchers and graduate students in
probability and statistics, especially those who wish to understand
the connections between Stein's method and Malliavin calculus.
Random Fields on the Sphere presents a comprehensive analysis of
isotropic spherical random fields. The main emphasis is on tools
from harmonic analysis, beginning with the representation theory
for the group of rotations SO(3). Many recent developments on the
method of moments and cumulants for the analysis of Gaussian
subordinated fields are reviewed. This background material is used
to analyse spectral representations of isotropic spherical random
fields and then to investigate in depth the properties of
associated harmonic coefficients. Properties and statistical
estimation of angular power spectra and polyspectra are addressed
in full. The authors are strongly motivated by cosmological
applications, especially the analysis of cosmic microwave
background (CMB) radiation data, which has initiated a challenging
new field of mathematical and statistical research. Ideal for
mathematicians and statisticians interested in applications to
cosmology, it will also interest cosmologists and mathematicians
working in group representations, stochastic calculus and spherical
wavelets.
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