This book defines and investigates the concept of a random object. To accomplish this task in a natural way, it brings together three major areas; statistical inference, measure-theoretic probability theory and stochastic processes. This point of view has not been explored by existing textbooks; one would need material on real analysis, measure and probability theory, as well as stochastic processes - in addition to at least one text on statistics- to capture the detail and depth of material that has gone into this volume.
Presents and illustrates 'random objects' in different contexts, under a unified framework, starting with rudimentary results on random variables and random sequences, all the way up to stochastic partial differential equations. Reviews rudimentary probability and introduces statistical inference, from basic to advanced, thus making the transition from basic statistical modeling and estimation to advanced topics more natural and concrete. Compact and comprehensive presentation of the material that will be useful to a reader from the mathematics and statistical sciences, at any stage of their career, either as a graduate student, an instructor, or an academician conducting research and requiring quick references and examples to classic topics. Includes 378 exercises, with the solutions manual available on the book's website. 121 illustrative examples of the concepts presented in the text (many including multiple items in a single example).
The book is targeted towards students at the master's and Ph.
D. levels, as well as, academicians in the mathematics, statistics and related disciplines. Basic knowledge of calculus and matrix algebra is required. Prior knowledge of probability or measure theory is welcomed but not necessary.
Rudimentary Models and Simulation Methods Statistical Inference Measure and Integration Theory Probability Theory Convergence of Random Objects Random Sequences Stochastic Processes Appendix
Athanasios Christou Micheas is Associate Professor of Statistics at the University of Missouri.
Reviews for Theory of Stochastic Objects: Probability, Stochastic Processes and Inference
This impressive text presents a wealth of material in probability and statistics, backed with measure theory, topology, and functional analysis. The ambition is to integrate, from the very beginning, probability with statistical ideas, frequentists and Bayesian, to build an understanding of why, and how to construct stochastic models. The target readers are students (and teachers) at masters and PhD level, as well as researchers in probability and statistics. The style is very clear and precise but quite compact, and the author gives essential advice on how to select material for courses at different levels and with different contents, based on many years of experience. Every chapter contains many exercises, some asking for proofs of important theorems, completed with computing and simulation experiments. -Georg Lindgren, Mathematical Statistics, Lund University