PROBABILITY_AND_ITS_APPLICA.PDF

Preface

Some thirty years ago it was still possible, as Loeve 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.