A collection of research level surveys on certain topics in probability theory by a well-known group of researchers. The book will be of interest to graduate students and researchers.

This Festschrift in honour of Paul Deheuvels’ 65th birthday compiles recent research results in the area between mathematical statistics and probability theory with a special emphasis on limit theorems. The book brings together contributions from invited international experts to provide an up-to-date survey of the field. Written in textbook style, this collection of original material addresses researchers, PhD and advanced Master students with a solid grasp of mathematical statistics and probability theory.

​Limit theorems and asymptotic results form a central topic in probability theory and mathematical statistics. New and non-classical limit theorems have been discovered for processes in random environments, especially in connection with random matrix theory and free probability. These questions and the techniques for answering them combine asymptotic enumerative combinatorics, particle systems and approximation theory, and are important for new approaches in geometric and metric number theory as well. Thus, the contributions in this book include a wide range of applications with surprising connections ranging from longest common subsequences for words, permutation groups, random matrices and free probability to entropy problems and metric number theory. The book is the product of a conference that took place in August 2011 in Bielefeld, Germany to celebrate the 60th birthday of Friedrich Götze, a noted expert in this field.

A leading authority sheds light on a variety of interesting topics in which probability theory plays a key role.

In this volume we study the value distribution of arithmetic functions, allowing unbounded renormalisations. The methods involve a synthesis of Probability and Number Theory; sums of independent infinitesimal random variables playing an important role. A central problem is to decide when an additive arithmetic function fin) admits a renormalisation by real functions a(x) and {3(x) > 0 so that asx ~ 00 the frequencies vx(n;f (n) - a(x) :s;; z {3 (x) ) converge weakly; (see Notation). In contrast to volume one we allow {3(x) to become unbounded with x. In particular, we investigate to what extent one can simulate the behaviour of additive arithmetic functions by that of sums of suit ably defined independent random variables. This fruiful point of view was intro duced in a 1939 paper of Erdos and Kac. We obtain their (now classical) result in Chapter 12. Subsequent methods involve both Fourier analysis on the line, and the appli cation of Dirichlet series. Many additional topics are considered. We mention only: a problem of Hardy and Ramanujan; local properties of additive arithmetic functions; the rate of convergence of certain arithmetic frequencies to the normal law; the arithmetic simulation of all stable laws. As in Volume I the historical background of various results is discussed, forming an integral part of the text. In Chapters 12 and 19 these considerations are quite extensive, and an author often speaks for himself.

This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications. Making this theorem an autonomous mathematical object was very important for the development of modern probability theory.

This book provides a comprehensive description of a new method of proving the central limit theorem, through the use of apparently unrelated results from information theory. It gives a basic introduction to the concepts of entropy and Fisher information, and collects together standard results concerning their behaviour. It brings together results from a number of research papers as well as unpublished material, showing how the techniques can give a unified view of limit theorems.

Martingale Limit Theory and Its Application discusses the asymptotic properties of martingales, particularly as regards key prototype of probabilistic behavior that has wide applications. The book explains the thesis that martingale theory is central to probability theory, and also examines the relationships between martingales and processes embeddable in or approximated by Brownian motion. The text reviews the martingale convergence theorem, the classical limit theory and analogs, and the martingale limit theorems viewed as the rate of convergence results in the martingale convergence theorem. The book explains the square function inequalities, weak law of large numbers, as well as the strong law of large numbers. The text discusses the reverse martingales, martingale tail sums, the invariance principles in the central limit theorem, and also the law of the iterated logarithm. The book investigates the limit theory for stationary processes via corresponding results for approximating martingales and the estimation of parameters from stochastic processes. The text can be profitably used as a reference for mathematicians, advanced students, and professors of higher mathematics or statistics.

This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The fourth edition begins with a short chapter on measure theory to orient readers new to the subject.

This treatise by an acknowledged expert includes several topics not found in any previous book.