Stationary Sequences And Random Fields
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Stationary Sequences and Random Fields
Author: Murray Rosenblatt
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
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.
Gaussian and Non-Gaussian Linear Time Series and Random Fields
Author: Murray Rosenblatt
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
Much of this book is concerned with autoregressive and moving av erage linear stationary sequences and random fields. These models are part of the classical literature in time series analysis, particularly in the Gaussian case. There is a large literature on probabilistic and statistical aspects of these models-to a great extent in the Gaussian context. In the Gaussian case best predictors are linear and there is an extensive study of the asymptotics of asymptotically optimal esti mators. Some discussion of these classical results is given to provide a contrast with what may occur in the non-Gaussian case. There the prediction problem may be nonlinear and problems of estima tion can have a certain complexity due to the richer structure that non-Gaussian models may have. Gaussian stationary sequences have a reversible probability struc ture, that is, the probability structure with time increasing in the usual manner is the same as that with time reversed. Chapter 1 considers the question of reversibility for linear stationary sequences and gives necessary and sufficient conditions for the reversibility. A neat result of Breidt and Davis on reversibility is presented. A sim ple but elegant result of Cheng is also given that specifies conditions for the identifiability of the filter coefficients that specify a linear non-Gaussian random field.
Random Fields and Geometry
Author: R. J. Adler
language: en
Publisher: Springer Science & Business Media
Release Date: 2009-01-29
This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. "Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory.