Random walk: A modern introduction.

*(English)*Zbl 1210.60002
Cambridge Studies in Advanced Mathematics 123. Cambridge: Cambridge University Press (ISBN 978-0-521-51918-2/hbk). xii, 364 p. (2010).

This is a contemporary introduction to the theory of random walks on the integer lattice that have increment distribution with zero mean and finite variance. In this situation the functional central limit theorem implies that the rescaled random walk converges to a Brownian motion. The central topic of this text are more precise error bounds for this convergence.

The first chapter introduces the three main classes of random walks considered in this text: symmetric finite range random walks, aperiodic random walks and the latter ones with zero mean and finite variance. In the second chapter the local central limit theorem is proved by using Fourier techniques. Then Brownian motion and the Skorokhod embedding are introduced. Other important topics are Green’s function and classical potential theory, applications of the local central limit theorem in dyadic coupling and finally as an outlook to current research connections between random walks, loop measures, spanning trees and determinants of Laplacians are established.

This book is a beautiful introduction to the theory of random walks for researchers as well as graduate students. It is assumed that the reader is familiar with basic real analysis, probability and measure theory. As a round-off the text includes a considerable number of exercises at the end of each chapter.

The first chapter introduces the three main classes of random walks considered in this text: symmetric finite range random walks, aperiodic random walks and the latter ones with zero mean and finite variance. In the second chapter the local central limit theorem is proved by using Fourier techniques. Then Brownian motion and the Skorokhod embedding are introduced. Other important topics are Green’s function and classical potential theory, applications of the local central limit theorem in dyadic coupling and finally as an outlook to current research connections between random walks, loop measures, spanning trees and determinants of Laplacians are established.

This book is a beautiful introduction to the theory of random walks for researchers as well as graduate students. It is assumed that the reader is familiar with basic real analysis, probability and measure theory. As a round-off the text includes a considerable number of exercises at the end of each chapter.

Reviewer: H. M. Mai (Berlin)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G50 | Sums of independent random variables; random walks |