This book, "Limit Theorems for Stochastic Processes" by Jean Jacod and Albert N. Shiryaev, is part of the "Grundlehren der mathematischen Wissenschaften" series. It focuses on the theory of weak convergence of probability measures on metric spaces, specifically for semimartingales. The authors provide a systematic exposition of the theory of convergence in law for stochastic processes that are semimartingales, which includes discrete-time processes, diffusions, Markov processes, point processes, and solutions of stochastic differential equations. The book is structured into several chapters, covering topics such as the general theory of stochastic processes, semimartingales, stochastic integrals, characteristics of semimartingales, martingale problems, changes of measures, Hellinger processes, absolute continuity and singularity of measures, contiguity, entire separation, convergence in variation, Skorokhod topology, and various limit theorems for processes with independent increments and semimartingales. Key aspects include: - **Semimartingales and Stochastic Integrals**: Definitions, properties, and constructions. - **Characteristics of Semimartingales**: Detailed exposition of the concept and its role in limit theorems. - **Martingale Problems**: Formulation and solutions for various processes. - **Contiguity and Absolute Continuity**: Analysis of measure contiguity and absolute continuity. - **Limit Theorems**: Conditions and results for convergence in law, including functional limit theorems and central limit theorems. The book also includes statistical applications, such as the convergence of likelihood ratio processes and the statistical invariance principle. It is a comprehensive resource for researchers and students interested in the theory and applications of limit theorems for stochastic processes.This book, "Limit Theorems for Stochastic Processes" by Jean Jacod and Albert N. Shiryaev, is part of the "Grundlehren der mathematischen Wissenschaften" series. It focuses on the theory of weak convergence of probability measures on metric spaces, specifically for semimartingales. The authors provide a systematic exposition of the theory of convergence in law for stochastic processes that are semimartingales, which includes discrete-time processes, diffusions, Markov processes, point processes, and solutions of stochastic differential equations. The book is structured into several chapters, covering topics such as the general theory of stochastic processes, semimartingales, stochastic integrals, characteristics of semimartingales, martingale problems, changes of measures, Hellinger processes, absolute continuity and singularity of measures, contiguity, entire separation, convergence in variation, Skorokhod topology, and various limit theorems for processes with independent increments and semimartingales. Key aspects include: - **Semimartingales and Stochastic Integrals**: Definitions, properties, and constructions. - **Characteristics of Semimartingales**: Detailed exposition of the concept and its role in limit theorems. - **Martingale Problems**: Formulation and solutions for various processes. - **Contiguity and Absolute Continuity**: Analysis of measure contiguity and absolute continuity. - **Limit Theorems**: Conditions and results for convergence in law, including functional limit theorems and central limit theorems. The book also includes statistical applications, such as the convergence of likelihood ratio processes and the statistical invariance principle. It is a comprehensive resource for researchers and students interested in the theory and applications of limit theorems for stochastic processes.