Foundations of Modern Probability - Olav Kallenberg
Some thirty years ago it was still possible, as Lo`eve so ably demonstrated,
to write a single book in probability theory containing practically everything
worth knowing in the subject. The subsequent development has been ex-plosive, and today a corresponding comprehensive coverage would require a
whole library. Researchers and graduate students alike seem compelled to a
rather extreme degree of specialization. As a result, the subject is threatened
by disintegration into dozens or hundreds of subfields.
At the same time the interaction between the areas is livelier than ever,
and there is a steadily growing core of key results and techniques that every
probabilist needs to know, if only to read the literature in his or her own
field. Thus, it seems essential that we all have at least a general overview of
the whole area, and we should do what we can to keep the subject together.
The present volume is an earnest attempt in that direction.
My original aim was to write a book about “everything.” Various space
and time constraints forced me to accept more modest and realistic goals
for the project. Thus, “foundations” had to be understood in the narrower
sense of the early 1970s, and there was no room for some of the more recent
developments. I especially regret the omission of topics such as large de-viations, Gibbs and Palm measures, interacting particle systems, stochastic
differential geometry, Malliavin calculus, SPDEs, measure-valued diffusions,
and branching and superprocesses. Clearly plenty of fundamental and in-triguing material remains for a possible second volume.
Even with my more limited, revised ambitions, I had to be extremely
selective in the choice of material. More importantly, it was necessary to look
for the most economical approach to every result I did decide to include. In
the latter respect, I was surprised to see how much could actually be done
to simplify and streamline proofs, often handed down through generations of
textbook writers. My general preference has been for results conveying some
new idea or relationship, whereas many propositions of a more technical
nature have been omitted. In the same vein, I have avoided technical or
computational proofs that give little insight into the proven results. This
conforms with my conviction that the logical structure is what matters most
in mathematics, even when applications is the ultimate goal.
Though the book is primarily intended as a general reference, it should
also be useful for graduate and seminar courses on different levels, ranging
from elementary to advanced. Thus, a first-year graduate course in measure-theoretic probability could be based on the first ten or so chapters, while
the rest of the book will readily provide material for more advanced courses
on various topics. Though the treatment is formally self-contained, as far
as measure theory and probability are concerned, the text is intended for
a rather sophisticated reader with at least some rudimentary knowledge of
subjects like topology, functional analysis, and complex variables.
My exposition is based on experiences from the numerous graduate and
seminar courses I have been privileged to teach in Sweden and in the United
States, ever since I was a graduate student myself. Over the years I have
developed a personal approach to almost every topic, and even experts might
find something of interest. Thus, many proofs may be new, and every chapter
contains results that are not available in the standard textbook literature. It
is my sincere hope that the book will convey some of the excitement I still
feel for the subject, which is without a doubt (even apart from its utter use-fulness) one of the richest and most beautiful areas of modern mathematics.
Notes and Acknowledgments: My first thanks are due to my numerous
Swedish teachers, and especially to Peter Jagers, whose 1971 seminar opened
my eyes to modern probability. The idea of this book was raised a few years
later when the analysts at Gothenburg asked me to give a short lecture course
on “probability for mathematicians.” Although I objected to the title, the
lectureswerepromptlydelivered, andIbecameconvincedoftheproject’sfea-sibility. For many years afterward I had a faithful and enthusiastic audience
in numerous courses on stochastic calculus, SDEs, and Markov processes. I
am grateful for that learning opportunity and for the feedback and encour-agement I received from colleagues and graduate students.
Inevitably I have benefited immensely from the heritage of countless au-thors, many of whom are not even listed in the bibliography. I have further
been fortunate to know many prominent probabilists of our time, who have
often inspired me through their scholarship and personal example. Two peo-ple, Klaus Matthes and Gopi Kallianpur, stand out as particularly important
influences in connection with my numerous visits to Berlin and Chapel Hill,
respectively.
The great Kai Lai Chung, my mentor and friend from recent years, offered
penetrating comments on all aspects of the work: linguistic, historical, and
mathematical. My colleague Ming Liao, always a stimulating partner for
discussions, was kind enough to check my material on potential theory. Early
versionsofthemanuscriptweretestedonseveralgroupsofgraduatestudents,
and Kamesh Casukhela, Davorin Dujmovic, and Hussain Talibi in particular
were helpful in spotting misprints. Ulrich Albrecht and Ed Slaminka offered
generous help with software problems. I am further grateful to John Kimmel,
Karina Mikhli, and the Springer production team for their patience with my
last-minute revisions and their truly professional handling of the project.
My greatest thanks go to my family, who is my constant source of happi-ness and inspiration. Without their love, encouragement, and understanding,
this work would not have been possible.
- pages: 535 pages
- Publisher: Springer 1997
- Language: English
- ISBN: 0-387-94957-7
Some thirty years ago it was still possible, as Lo`eve so ably demonstrated,
to write a single book in probability theory containing practically everything
worth knowing in the subject. The subsequent development has been ex-plosive, and today a corresponding comprehensive coverage would require a
whole library. Researchers and graduate students alike seem compelled to a
rather extreme degree of specialization. As a result, the subject is threatened
by disintegration into dozens or hundreds of subfields.
At the same time the interaction between the areas is livelier than ever,
and there is a steadily growing core of key results and techniques that every
probabilist needs to know, if only to read the literature in his or her own
field. Thus, it seems essential that we all have at least a general overview of
the whole area, and we should do what we can to keep the subject together.
The present volume is an earnest attempt in that direction.
My original aim was to write a book about “everything.” Various space
and time constraints forced me to accept more modest and realistic goals
for the project. Thus, “foundations” had to be understood in the narrower
sense of the early 1970s, and there was no room for some of the more recent
developments. I especially regret the omission of topics such as large de-viations, Gibbs and Palm measures, interacting particle systems, stochastic
differential geometry, Malliavin calculus, SPDEs, measure-valued diffusions,
and branching and superprocesses. Clearly plenty of fundamental and in-triguing material remains for a possible second volume.
Even with my more limited, revised ambitions, I had to be extremely
selective in the choice of material. More importantly, it was necessary to look
for the most economical approach to every result I did decide to include. In
the latter respect, I was surprised to see how much could actually be done
to simplify and streamline proofs, often handed down through generations of
textbook writers. My general preference has been for results conveying some
new idea or relationship, whereas many propositions of a more technical
nature have been omitted. In the same vein, I have avoided technical or
computational proofs that give little insight into the proven results. This
conforms with my conviction that the logical structure is what matters most
in mathematics, even when applications is the ultimate goal.
Though the book is primarily intended as a general reference, it should
also be useful for graduate and seminar courses on different levels, ranging
from elementary to advanced. Thus, a first-year graduate course in measure-theoretic probability could be based on the first ten or so chapters, while
the rest of the book will readily provide material for more advanced courses
on various topics. Though the treatment is formally self-contained, as far
as measure theory and probability are concerned, the text is intended for
a rather sophisticated reader with at least some rudimentary knowledge of
subjects like topology, functional analysis, and complex variables.
My exposition is based on experiences from the numerous graduate and
seminar courses I have been privileged to teach in Sweden and in the United
States, ever since I was a graduate student myself. Over the years I have
developed a personal approach to almost every topic, and even experts might
find something of interest. Thus, many proofs may be new, and every chapter
contains results that are not available in the standard textbook literature. It
is my sincere hope that the book will convey some of the excitement I still
feel for the subject, which is without a doubt (even apart from its utter use-fulness) one of the richest and most beautiful areas of modern mathematics.
Notes and Acknowledgments: My first thanks are due to my numerous
Swedish teachers, and especially to Peter Jagers, whose 1971 seminar opened
my eyes to modern probability. The idea of this book was raised a few years
later when the analysts at Gothenburg asked me to give a short lecture course
on “probability for mathematicians.” Although I objected to the title, the
lectureswerepromptlydelivered, andIbecameconvincedoftheproject’sfea-sibility. For many years afterward I had a faithful and enthusiastic audience
in numerous courses on stochastic calculus, SDEs, and Markov processes. I
am grateful for that learning opportunity and for the feedback and encour-agement I received from colleagues and graduate students.
Inevitably I have benefited immensely from the heritage of countless au-thors, many of whom are not even listed in the bibliography. I have further
been fortunate to know many prominent probabilists of our time, who have
often inspired me through their scholarship and personal example. Two peo-ple, Klaus Matthes and Gopi Kallianpur, stand out as particularly important
influences in connection with my numerous visits to Berlin and Chapel Hill,
respectively.
The great Kai Lai Chung, my mentor and friend from recent years, offered
penetrating comments on all aspects of the work: linguistic, historical, and
mathematical. My colleague Ming Liao, always a stimulating partner for
discussions, was kind enough to check my material on potential theory. Early
versionsofthemanuscriptweretestedonseveralgroupsofgraduatestudents,
and Kamesh Casukhela, Davorin Dujmovic, and Hussain Talibi in particular
were helpful in spotting misprints. Ulrich Albrecht and Ed Slaminka offered
generous help with software problems. I am further grateful to John Kimmel,
Karina Mikhli, and the Springer production team for their patience with my
last-minute revisions and their truly professional handling of the project.
My greatest thanks go to my family, who is my constant source of happi-ness and inspiration. Without their love, encouragement, and understanding,
this work would not have been possible.
