Krishna B. Athreya – författare

This book arose out of two graduate courses that the authors have taught duringthepastseveralyears;the?rstonebeingonmeasuretheoryfollowed by the second one on advanced probability theory. The traditional approach to a ?rst course in measure theory, such as in Royden (1988), is to teach the Lebesgue measure on the real line, then the p di?erentation theorems of Lebesgue, L -spaces on R, and do general m- sure at the end of the course with one main application to the construction of product measures. This approach does have the pedagogic advantage of seeing one concrete case ?rst before going to the general one. But this also has the disadvantage in making many students’ perspective on m- sure theory somewhat narrow. It leads them to think only in terms of the Lebesgue measure on the real line and to believe that measure theory is intimately tied to the topology of the real line. As students of statistics, probability, physics, engineering, economics, and biology know very well, there are mass distributions that are typically nonuniform, and hence it is useful to gain a general perspective. This book attempts to provide that general perspective right from the beginning. The opening chapter gives an informal introduction to measure and integration theory. It shows that the notions of ?-algebra of sets and countable additivity of a set function are dictated by certain very na- ral approximation procedures from practical applications and that they are not just some abstract ideas.

This book arose out of two graduate courses that the authors have taught duringthepastseveralyears;the?rstonebeingonmeasuretheoryfollowed by the second one on advanced probability theory. The traditional approach to a ?rst course in measure theory, such as in Royden (1988), is to teach the Lebesgue measure on the real line, then the p di?erentation theorems of Lebesgue, L -spaces on R, and do general m- sure at the end of the course with one main application to the construction of product measures. This approach does have the pedagogic advantage of seeing one concrete case ?rst before going to the general one. But this also has the disadvantage in making many students’ perspective on m- sure theory somewhat narrow. It leads them to think only in terms of the Lebesgue measure on the real line and to believe that measure theory is intimately tied to the topology of the real line. As students of statistics, probability, physics, engineering, economics, and biology know very well, there are mass distributions that are typically nonuniform, and hence it is useful to gain a general perspective. This book attempts to provide that general perspective right from the beginning. The opening chapter gives an informal introduction to measure and integration theory. It shows that the notions of ?-algebra of sets and countable additivity of a set function are dictated by certain very na- ral approximation procedures from practical applications and that they are not just some abstract ideas.

This IMA Volume in Mathematics and its Applications CLASSICAL AND MODERN BRANCHING PROCESSES is based on the proceedings with the same title and was an integral part of the 1993-94 IMA program on "Emerging Applications of Probability." We would like to thank Krishna B. Athreya and Peter J agers for their hard work in organizing this meeting and in editing the proceedings. We also take this opportunity to thank the National Science Foundation, the Army Research Office, and the National Security Agency, whose financial support made this workshop possible. A vner Friedman Robert Gulliver v PREFACE The IMA workshop on Classical and Modern Branching Processes was held during June 13-171994 as part of the IMA year on Emerging Appli­ cations of Probability. The organizers of the year long program identified branching processes as one of the active areas in which a workshop should be held. Krish­ na B. Athreya and Peter Jagers were asked to organize this. The topics covered by the workshop could broadly be divided into the following areas: 1. Tree structures and branching processes; 2. Branching random walks; 3. Measure valued branching processes; 4. Branching with dependence; 5. Large deviations in branching processes; 6. Classical branching processes.

This book arose out of two graduate courses that the authors have taught duringthepastseveralyears;the?rstonebeingonmeasuretheoryfollowed by the second one on advanced probability theory. The traditional approach to a ?rst course in measure theory, such as in Royden (1988), is to teach the Lebesgue measure on the real line, then the p di?erentation theorems of Lebesgue, L -spaces on R, and do general m- sure at the end of the course with one main application to the construction of product measures. This approach does have the pedagogic advantage of seeing one concrete case ?rst before going to the general one. But this also has the disadvantage in making many students’ perspective on m- sure theory somewhat narrow. It leads them to think only in terms of the Lebesgue measure on the real line and to believe that measure theory is intimately tied to the topology of the real line. As students of statistics, probability, physics, engineering, economics, and biology know very well, there are mass distributions that are typically nonuniform, and hence it is useful to gain a general perspective. This book attempts to provide that general perspective right from the beginning. The opening chapter gives an informal introduction to measure and integration theory. It shows that the notions of ?-algebra of sets and countable additivity of a set function are dictated by certain very na- ral approximation procedures from practical applications and that they are not just some abstract ideas.

This IMA Volume in Mathematics and its Applications CLASSICAL AND MODERN BRANCHING PROCESSES is based on the proceedings with the same title and was an integral part of the 1993-94 IMA program on "Emerging Applications of Probability." We would like to thank Krishna B. Athreya and Peter J agers for their hard work in organizing this meeting and in editing the proceedings. We also take this opportunity to thank the National Science Foundation, the Army Research Office, and the National Security Agency, whose financial support made this workshop possible. A vner Friedman Robert Gulliver v PREFACE The IMA workshop on Classical and Modern Branching Processes was held during June 13-171994 as part of the IMA year on Emerging Appli­ cations of Probability. The organizers of the year long program identified branching processes as one of the active areas in which a workshop should be held. Krish­ na B. Athreya and Peter Jagers were asked to organize this. The topics covered by the workshop could broadly be divided into the following areas: 1. Tree structures and branching processes; 2. Branching random walks; 3. Measure valued branching processes; 4. Branching with dependence; 5. Large deviations in branching processes; 6. Classical branching processes.

This IMA Volume in Mathematics and its Applications CLASSICAL AND MODERN BRANCHING PROCESSES is based on the proceedings with the same title and was an integral part of the 1993-94 IMA program on "Emerging Applications of Probability." We would like to thank Krishna B. Athreya and Peter J agers for their hard work in organizing this meeting and in editing the proceedings. We also take this opportunity to thank the National Science Foundation, the Army Research Office, and the National Security Agency, whose financial support made this workshop possible. A vner Friedman Robert Gulliver v PREFACE The IMA workshop on Classical and Modern Branching Processes was held during June 13-171994 as part of the IMA year on Emerging Appli­ cations of Probability. The organizers of the year long program identified branching processes as one of the active areas in which a workshop should be held. Krish­ na B. Athreya and Peter Jagers were asked to organize this. The topics covered by the workshop could broadly be divided into the following areas: 1. Tree structures and branching processes; 2. Branching random walks; 3. Measure valued branching processes; 4. Branching with dependence; 5. Large deviations in branching processes; 6. Classical branching processes.

Probability, Statistics, and Mathematics: Papers in Honor of Samuel Karlin is a collection of papers dealing with probability, statistics, and mathematics. Conceived in honor of Polish-born mathematician Samuel Karlin, the book covers a wide array of topics, from the second-order moments of a stationary Markov chain to the exponentiality of the local time at hitting times for reflecting diffusions. Smoothed limit theorems for equilibrium processes are also discussed. Comprised of 24 chapters, this book begins with an introduction to the second-order moments of a stationary Markov chain, paying particular attention to the consequences of the autoregressive structure of the vector-valued process and how to estimate the stationary probabilities from a finite sequence of observations. Subsequent chapters focus on A. Selberg''s second beta integral and an integral of mehta; a normal approximation for the number of local maxima of a random function on a graph; nonnegative polynomials on polyhedra; and the fundamental period of the queue with Markov-modulated arrivals. The rate of escape problem for a class of random walks is also considered. This monograph is intended for students and practitioners in the fields of statistics, mathematics, and economics.

The purpose of this book is to give a unified treatment of the limit theory of branching processes. Since the publication of the important book of T E. Harris (Theory of Branching Processes, Springer, 1963) the subject has developed and matured significantly. Many of the classical limit laws are now known in their sharpest form, and there are new proofs that give insight into the results. Our work deals primarily with this decade, and thus has very little overlap with that of Harris. Only enough material is repeated to make the treatment essentially self-contained. For example, certain foundational questions on the construction of processes, to which we have nothing new to add, are not developed. There is a natural classification of branching processes according to their criticality condition, their time parameter, the single or multi-type particle cases, the Markovian or non-Markovian character of the pro­ cess, etc. We have tried to avoid the rather uneconomical and un­ enlightening approach of treating these categories independently, and by a series of similar but increasingly complicated techniques. The basic Galton-Watson process is developed in great detail in Chapters I and II.