Murray Rosenblatt - Böcker

The Athens Conference on Applied Probability and Time Series in 1995 brought together researchers from across the world. The published papers appear in two volumes. Volume II presents papers on time series analysis, many of which were contributed to a meeting in March 1995 partly in honour of E.J. Hannan. The initial paper by P.M. Robinson discusses Ted Hannan's researches and their influence on current work in time series analysis. Other papers discuss methods for finite parameter Gaussian models, time series with infinite variance or stable marginal distribution, frequency domain methods, long range dependent processes, nonstationary processes, and nonlinear time series. The methods presented can be applied in a number of fields such as statistics, applied mathematics, engineering, economics and ecology. The papers include many of the topics of current interest in time series analysis and will be of interest to a wide range of researchers.

The book is concerned with linear time series and random fields in both the Gaussian and especially the non-Gaussian context. The principal focus is on autoregressive moving average models and analogous random fields. Probabilistic and statistical questions are both discussed. The Gaussian models are contrasted with noncausal or noninvertible (nonminimum phase) non-Gaussian models which can have a much richer structure than Gaussian models. The book deals with problems of prediction (which can have a nonlinear character) and estimation. New results for nonminimum phase non-Gaussian processes are exposited and open questions are noted. The book is intended as a text for graduate students in statistics, mathematics, engineering, the natural sciences and economics. An initial background in probability theory and statistics is suggested. Notes on background, history and open problems are given at the end of the book. Murray Rosenblatt is Professor of Mathematics at the University of California, San Diego. He was a Guggenheim Fellow in 1965 and 1972 and is a member of the National Academy of Sciences, U.S.A.He is the author of Random Processes (1962), Markov Processes: Structure and Asymptotic Behavior (1971), Stationary Sequences and Random Fields (1985), and Stochastic Curve Estimation (1991).

This book has a dual purpose. One of these is to present material which selec­ tively will be appropriate for a quarter or semester course in time series analysis and which will cover both the finite parameter and spectral approach. The second object is the presentation of topics of current research interest and some open questions. I mention these now. In particular, there is a discussion in Chapter III of the types of limit theorems that will imply asymptotic nor­ mality for covariance estimates and smoothings of the periodogram. This dis­ cussion allows one to get results on the asymptotic distribution of finite para­ meter estimates that are broader than those usually given in the literature in Chapter IV. A derivation of the asymptotic distribution for spectral (second order) estimates is given under an assumption of strong mixing in Chapter V. A discussion of higher order cumulant spectra and their large sample properties under appropriate moment conditions follows in Chapter VI. Probability density, conditional probability density and regression estimates are considered in Chapter VII under conditions of short range dependence. Chapter VIII deals with a number of topics. At first estimates for the structure function of a large class of non-Gaussian linear processes are constructed. One can determine much more about this structure or transfer function in the non-Gaussian case than one can for Gaussian processes. In particular, one can determine almost all the phase information.

From the Preface to the First Edition (1957): The purpose of this book is two-fold. It is written in the terminology of the theoretical statistician because one of our objectives is to direct his attention to an approach to time series analysis that is essentially different from most of the techniques used by time series analysts in the past. The second objective is to present a unified treatment of methods that are being used increasingly in the physical sciences and technology. We hope that the book will be of considerable interest to research workers in these fields. Keeping the first objective in mind, we have given a rigorous mathematical discussion of these new topics in time series analysis.The existing literature in time series analysis is characterized with few exceptions by a lack of precision both in conception and in the mathematical treatment of the problems dealt with. To avoid this vagueness, we have devoted more space to rigorous proofs than may appear necessary to some readers, but we believe that a study of the proofs will furnish valuable clues to the practical validity of the results and be an important guide to intuition. We have tried to balance the formal proofs with intuitive remarks and comments on practical applications. While the regularity assumptions we have required in many cases may seem restrictive, appropriately interpreted they give an indication of the range in which the methods are practically valid. We have made such interpretations in the comments accompanying the formal proofs. Readers should have knowledge of statistics and basic probability. The second edition was printed with corrections.

This collection of six articles provides the reader with a range of ideas and applications of probability theory. Written by some of leading experts in the field, the articles touch upon such topics as sequential methods in statistical inference, ergodicity and mixing for stationary random processes, distribution of maxima of independent sequences, asymptotic analysis of stochastic differential equations, mathematical aspects of statistical mechanics, and the use of entropy in the problem of isomorphism of ergodic dynamical systems.

Much of this book is concerned with autoregressive and moving av­ erage linear stationary sequences and random fields. Chapter 1 considers the question of reversibility for linear stationary sequences and gives necessary and sufficient conditions for the reversibility.

Part of a two volume set based on a recent IMA program of the same name. The goal of the program and these books is to develop a community of statistical and other scientists kept up-to-date on developments in this quickly evolving and interdisciplinary field. Consequently, these books present recent material by distinguished researchers. Topics discussed in Part I include nonlinear and non- Gaussian models and processes (higher order moments and spectra, nonlinear systems, applications in astronomy, geophysics, engineering, and simulation) and the interaction of time series analysis and statistics (information model identification, categorical valued time series, nonparametric and semiparametric methods). Self-similar processes and long-range dependence (time series with long memory, fractals, 1/f noise, stable noise) and time series research common to engineers and economists (modeling of multivariate and possibly non-stationary time series, state space and adaptive methods) are discussed in Part II.

This IMA Volume in Mathematics and its Applications NEW DIRECTIONS IN TIME SERIES ANALYSIS, PART II is based on the proceedings of the IMA summer program "New Directions in Time Series Analysis. " We are grateful to David Brillinger, Peter Caines, John Geweke, Emanuel Parzen, Murray Rosenblatt, and Murad Taqqu for organizing the program and we hope that the remarkable excitement and enthusiasm of the participants in this interdisciplinary effort are communicated to the reader. A vner Friedman Willard Miller, Jr. PREFACE Time Series Analysis is truly an interdisciplinary field because development of its theory and methods requires interaction between the diverse disciplines in which it is applied. To harness its great potential, strong interaction must be encouraged among the diverse community of statisticians and other scientists whose research involves the analysis of time series data. This was the goal of the IMA Workshop on "New Directions in Time Series Analysis. " The workshop was held July 2-July 27, 1990 and was organized by a committee consisting of Emanuel Parzen (chair), David Brillinger, Murray Rosenblatt, Murad S. Taqqu, John Geweke, and Peter Caines. Constant guidance and encouragement was provided by Avner Friedman, Director of the IMA, and his very helpful and efficient staff. The workshops were organized by weeks. It may be of interest to record the themes that were announced in the IMA newsletter describing the workshop: l.