Measure Theoretic Probability

Applies the well-developed tools of the theory of weak convergence of probability measures to large deviation analysis–a consistent new approach
The theory of large deviations, one of the most dynamic topics in probability today, studies rare events in stochastic systems. The nonlinear nature of the theory contributes both to its richness and difficulty. This innovative text demonstrates how to employ the well-established linear techniques of weak convergence theory to prove large deviation results. Beginning with a step-by-step development of the approach, the book skillfully guides readers through models of increasing complexity covering a wide variety of random variable-level and process-level problems. Representation formulas for large deviation-type expectations are a key tool and are developed systematically for discrete-time problems.

Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. In a lively and imaginative presentation, studded with examples, exercises, and applications, and supported by inclusion of computational procedures, the author has created a textbook that provides easy access to this fundamental topic for many students of applied sciences at many levels. With its carefully modularized discussion and crystal clear differentiation between rigorous proof and plausibility argument, it is accessible to beginners but flexible enough to serve as well those who come to the course with strong backgrounds.

Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference tool with broad appeal as well as a platform for future research. Fourteen chapters are organized into three parts: mathematical logic and foundations (Chapters 1-2), general topology (Chapters 3-10), and measure and probability theory (Chapters 11-14).