Preface
This book is based on two four-hour courses on advanced probability theory that I
have held in recent years at the universities of Cologne and Mainz. It is implicitly as-
sumed that the reader has a certain familiarity with the basic concepts of probability
theory, although the formal framework will be fully developed in this book.
The aim of this book is to present the central objects and concepts of probability
theory: random variables, independence, laws of large numbers and central limit
theorems, martingales, exchangeability and infinite divisibility, Markov chains and
Markov processes, as well as their connection with discrete potential theory, cou-
pling, ergodic theory, Brownian motion and the It
ˆ
o integral (including stochastic
differential equations), the Poisson point process, percolation and the theory of large
deviations.
Measure theory and integration are necessary prerequisites for a systematic proba-
bility theory. We develop it only to the point to which it is needed for our purposes:
construction of measures and integrals, the Radon-Nikodym theorem and regular
conditional distributions, convergence theorems for functions (Lebesgue) and mea-
sures (Prohorov) and construction of measures in product spaces. The chapters on
measure theory do not come as a block at the beginning (although they are written
such that this would be possible; that is, independent of the probabilistic chapters)
but are rather interlaced with probabilistic chapters that are designed to display the
power of the abstract concepts in the more intuitive world of probability theory. For
example, we study percolation theory at the point where we barely have measures,
random variables and independence; not even the integral is needed. As the only ex-
ception, the systematic construction of independent random variables is deferred to
Chapter 14. Although it is rather a matter of taste, I hope that this setup helps to
motivate the reader throughout the measure-theoretical chapters.
Those readers with a solid measure-theoretical education can skip in particular the
first and fourth chapters and might wish only to look up this or that.
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