The book "Stable Non-Gaussian Random Processes" by Gennady Samorodnitsky and Murad S. Taqqu provides a comprehensive treatment of stable non-Gaussian random processes, which are essential in modeling high variability and heavy-tailed phenomena. The authors emphasize the probabilistic approach, focusing on tails, moments, and dependence structures, and cover both univariate and multivariate stable distributions. Key topics include: 1. **Univariate Stable Distributions**: The book introduces stable distributions through four equivalent definitions, emphasizing their role in the central limit theorem and their infinite variance property. It discusses symmetric and skewed stable distributions, their characteristic functions, and series representations. 2. **Multivariate Stable Distributions**: The text explores multivariate stable distributions, including strictly stable random vectors and the concept of spectral measure. It highlights the differences between Gaussian and non-Gaussian stable distributions, particularly in terms of tail behavior and skewness. 3. **Stable Stochastic Integrals and Processes**: The book delves into the theory of stable stochastic integrals and processes, providing multiple definitions and constructive methods for their representation. It covers sub-Gaussian and sub-stable processes, as well as the connection between stable processes and fractional Brownian motion. 4. **Dependence Structures**: The authors study the dependence structures of multivariate stable distributions, including conditioning, order statistics, joint moments, and codifference. They also analyze regression in detail, both linear and non-linear, and provide exact forms for regression functions. 5. **Harmonizable Processes**: The book discusses harmonizable stable processes and their representation, which is crucial for understanding self-similar processes and their applications in various fields such as geophysics, hydrology, and economics. 6. **Self-Similar Processes**: The text examines self-similar processes, also known as "random fractals," and their properties, including long-range dependence and scaling invariance. It covers fractional Brownian motions and stable processes, and discusses ARMA and fractional ARIMA models with stable innovations. 7. **Chentsov Random Fields**: The authors introduce Chentsov random fields and extend the geometric construction of Lévy Brownian motion to create a variety of self-similar stable random fields. 8. **Sample Path Properties**: The book explores the sample path properties of stable processes, including boundedness, continuity, and oscillations. It provides detailed analysis of these properties and their implications for practical applications. 9. **Integral Representation**: The text concludes with a detailed discussion of integral representations of α-stable processes, offering a comprehensive understanding of their structure and behavior. The book is designed to be accessible to researchers and graduate students in probability, applied probability, and statistics, providing a thorough introduction to the theory and applications of stable non-Gaussian random processes.The book "Stable Non-Gaussian Random Processes" by Gennady Samorodnitsky and Murad S. Taqqu provides a comprehensive treatment of stable non-Gaussian random processes, which are essential in modeling high variability and heavy-tailed phenomena. The authors emphasize the probabilistic approach, focusing on tails, moments, and dependence structures, and cover both univariate and multivariate stable distributions. Key topics include: 1. **Univariate Stable Distributions**: The book introduces stable distributions through four equivalent definitions, emphasizing their role in the central limit theorem and their infinite variance property. It discusses symmetric and skewed stable distributions, their characteristic functions, and series representations. 2. **Multivariate Stable Distributions**: The text explores multivariate stable distributions, including strictly stable random vectors and the concept of spectral measure. It highlights the differences between Gaussian and non-Gaussian stable distributions, particularly in terms of tail behavior and skewness. 3. **Stable Stochastic Integrals and Processes**: The book delves into the theory of stable stochastic integrals and processes, providing multiple definitions and constructive methods for their representation. It covers sub-Gaussian and sub-stable processes, as well as the connection between stable processes and fractional Brownian motion. 4. **Dependence Structures**: The authors study the dependence structures of multivariate stable distributions, including conditioning, order statistics, joint moments, and codifference. They also analyze regression in detail, both linear and non-linear, and provide exact forms for regression functions. 5. **Harmonizable Processes**: The book discusses harmonizable stable processes and their representation, which is crucial for understanding self-similar processes and their applications in various fields such as geophysics, hydrology, and economics. 6. **Self-Similar Processes**: The text examines self-similar processes, also known as "random fractals," and their properties, including long-range dependence and scaling invariance. It covers fractional Brownian motions and stable processes, and discusses ARMA and fractional ARIMA models with stable innovations. 7. **Chentsov Random Fields**: The authors introduce Chentsov random fields and extend the geometric construction of Lévy Brownian motion to create a variety of self-similar stable random fields. 8. **Sample Path Properties**: The book explores the sample path properties of stable processes, including boundedness, continuity, and oscillations. It provides detailed analysis of these properties and their implications for practical applications. 9. **Integral Representation**: The text concludes with a detailed discussion of integral representations of α-stable processes, offering a comprehensive understanding of their structure and behavior. The book is designed to be accessible to researchers and graduate students in probability, applied probability, and statistics, providing a thorough introduction to the theory and applications of stable non-Gaussian random processes.