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The title High Dimensional Probability is an attempt to describe
the many trib utaries of research on Gaussian processes and
probability in Banach spaces that started in the early 1970's. In
each of these fields it is necessary to consider large classes of
stochastic processes under minimal conditions. There are rewards in
re search of this sort. One can often gain deep insights, even
about familiar processes, by stripping away details that in
hindsight turn out to be extraneous. Many of the problems that
motivated researchers in the 1970's were solved. But the powerful
new tools created for their solution, such as randomization,
isoperimetry, concentration of measure, moment and exponential
inequalities, chaining, series representations and decoupling
turned out to be applicable to other important areas of
probability. They led to significant advances in the study of
empirical processes and other topics in theoretical statistics and
to a new ap proach to the study of aspects of Levy processes and
Markov processes in general. Papers on these topics as well as on
the continuing study of Gaussian processes and probability in
Banach are included in this volume."
The papers contained in this volume are an indication of the topics
th discussed and the interests of the participants of The 9
International Conference on Probability in Banach Spaces, held at
Sandjberg, Denmark, August 16-21, 1993. A glance at the table of
contents indicates the broad range of topics covered at this
conference. What defines research in this field is not so much the
topics considered but the generality of the ques tions that are
asked. The goal is to examine the behavior of large classes of
stochastic processes and to describe it in terms of a few simple
prop erties that the processes share. The reward of research like
this is that occasionally one can gain deep insight, even about
familiar processes, by stripping away details, that in hindsight
turn out to be extraneous. A good understanding about the
disciplines involved in this field can be obtained from the recent
book, Probability in Banach Spaces, Springer-Verlag, by M. Ledoux
and M. Thlagrand. On page 5, of this book, there is a list of
previous conferences in probability in Banach spaces, including the
other eight international conferences. One can see that research in
this field over the last twenty years has contributed significantly
to knowledge in probability and has had important applications in
many other branches of mathematics, most notably in statistics and
functional analysis."
The title High Dimensional Probability is used to describe the
many tributaries of research on Gaussian processes and probability
in Banach spaces that started in the early 1970s. Many of the
problems that motivated researchers at that time were solved. But
the powerful new tools created for their solution turned out to be
applicable to other important areas of probability. They led to
significant advances in the study of empirical processes and other
topics in theoretical statistics and to a new approach to the study
of aspects of Levy processes and Markov processes in general. The
papers in this book reflect these broad categories. The volume thus
will be a valuable resource for postgraduates and reseachers in
probability theory and mathematical statistics."
The papers contained in this volume are an indication of the topics
th discussed and the interests of the participants of The 9
International Conference on Probability in Banach Spaces, held at
Sandjberg, Denmark, August 16-21, 1993. A glance at the table of
contents indicates the broad range of topics covered at this
conference. What defines research in this field is not so much the
topics considered but the generality of the ques tions that are
asked. The goal is to examine the behavior of large classes of
stochastic processes and to describe it in terms of a few simple
prop erties that the processes share. The reward of research like
this is that occasionally one can gain deep insight, even about
familiar processes, by stripping away details, that in hindsight
turn out to be extraneous. A good understanding about the
disciplines involved in this field can be obtained from the recent
book, Probability in Banach Spaces, Springer-Verlag, by M. Ledoux
and M. Thlagrand. On page 5, of this book, there is a list of
previous conferences in probability in Banach spaces, including the
other eight international conferences. One can see that research in
this field over the last twenty years has contributed significantly
to knowledge in probability and has had important applications in
many other branches of mathematics, most notably in statistics and
functional analysis."
This SpringerBriefs employs a novel approach to obtain the precise
asymptotic behavior at infinity of a large class of permanental
sequences related to birth and death processes and autoregressive
Gaussian sequences using techniques from the theory of Gaussian
processes and Markov chains. The authors study alpha-permanental
processes that are positive infinitely divisible processes
determined by the potential density of a transient Markov process.
When the Markov process is symmetric, a 1/2-permanental process is
the square of a Gaussian process. Permanental processes are related
by the Dynkin isomorphism theorem to the total accumulated local
time of the Markov process when the potential density is symmetric,
and by a generalization of the Dynkin theorem by Eisenbaum and
Kaspi without requiring symmetry. Permanental processes are also
related to chi square processes and loop soups. The book appeals to
researchers and advanced graduate students interested in stochastic
processes, infinitely divisible processes and Markov chains.
This book was first published in 2006. Written by two of the
foremost researchers in the field, this book studies the local
times of Markov processes by employing isomorphism theorems that
relate them to certain associated Gaussian processes. It builds to
this material through self-contained but harmonized 'mini-courses'
on the relevant ingredients, which assume only knowledge of
measure-theoretic probability. The streamlined selection of topics
creates an easy entrance for students and experts in related
fields. The book starts by developing the fundamentals of Markov
process theory and then of Gaussian process theory, including
sample path properties. It then proceeds to more advanced results,
bringing the reader to the heart of contemporary research. It
presents the remarkable isomorphism theorems of Dynkin and
Eisenbaum and then shows how they can be applied to obtain new
properties of Markov processes by using well-established techniques
in Gaussian process theory. This original, readable book will
appeal to both researchers and advanced graduate students.
In this book the authors give the first necessary and sufficient
conditions for the uniform convergence a.s. of random Fourier
series on locally compact Abelian groups and on compact non-Abelian
groups. They also obtain many related results. For example,
whenever a random Fourier series converges uniformly a.s. it also
satisfies the central limit theorem. The methods developed are used
to study some questions in harmonic analysis that are not
intrinsically random. For example, a new characterization of Sidon
sets is derived. The major results depend heavily on the
Dudley-Fernique necessary and sufficient condition for the
continuity of stationary Gaussian processes and on recent work on
sums of independent Banach space valued random variables. It is
noteworthy that the proofs for the Abelian case immediately extend
to the non-Abelian case once the proper definition of random
Fourier series is made. In doing this the authors obtain new
results on sums of independent random matrices with elements in a
Banach space. The final chapter of the book suggests several
directions for further research.
This book was first published in 2006. Written by two of the
foremost researchers in the field, this book studies the local
times of Markov processes by employing isomorphism theorems that
relate them to certain associated Gaussian processes. It builds to
this material through self-contained but harmonized 'mini-courses'
on the relevant ingredients, which assume only knowledge of
measure-theoretic probability. The streamlined selection of topics
creates an easy entrance for students and experts in related
fields. The book starts by developing the fundamentals of Markov
process theory and then of Gaussian process theory, including
sample path properties. It then proceeds to more advanced results,
bringing the reader to the heart of contemporary research. It
presents the remarkable isomorphism theorems of Dynkin and
Eisenbaum and then shows how they can be applied to obtain new
properties of Markov processes by using well-established techniques
in Gaussian process theory. This original, readable book will
appeal to both researchers and advanced graduate students.
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