This book, authored by Edgar Alirio Valencia Angulo, explores the use of embedding distributions into reproducing kernel Hilbert spaces (RKHS) to develop metrics between random processes. The primary focus is on the development and application of these metrics in time series classification and the estimation of autoregressive models. The book is structured into four chapters: 1. **Introduction**: Introduces the background and motivation for using RKHS embeddings in the context of random processes, highlighting the advantages and potential applications. 2. **RKHS and Tensor Product RKHS**: Discusses the theoretical foundations of RKHS, including definitions, properties, and examples. It also introduces the concept of tensor products of RKHSs and their applications. 3. **Metrics Between Probability Distributions**: Explores various metrics between probability distributions, emphasizing the use of Parzen estimators and specific kernels like Gaussian and Laplacian. The chapter provides analytical expressions for new metrics and discusses their properties. 4. **Metrics Between Random Processes**:Applies the metrics developed in the previous chapters to random processes, particularly Hidden Markov Models (HMMs) and autoregressive processes. It includes experimental results and demonstrations of how these metrics can be used for time series prediction and model comparison. The book aims to provide a comprehensive framework for understanding and applying RKHS embeddings in the context of random processes, offering both theoretical insights and practical applications.This book, authored by Edgar Alirio Valencia Angulo, explores the use of embedding distributions into reproducing kernel Hilbert spaces (RKHS) to develop metrics between random processes. The primary focus is on the development and application of these metrics in time series classification and the estimation of autoregressive models. The book is structured into four chapters: 1. **Introduction**: Introduces the background and motivation for using RKHS embeddings in the context of random processes, highlighting the advantages and potential applications. 2. **RKHS and Tensor Product RKHS**: Discusses the theoretical foundations of RKHS, including definitions, properties, and examples. It also introduces the concept of tensor products of RKHSs and their applications. 3. **Metrics Between Probability Distributions**: Explores various metrics between probability distributions, emphasizing the use of Parzen estimators and specific kernels like Gaussian and Laplacian. The chapter provides analytical expressions for new metrics and discusses their properties. 4. **Metrics Between Random Processes**:Applies the metrics developed in the previous chapters to random processes, particularly Hidden Markov Models (HMMs) and autoregressive processes. It includes experimental results and demonstrations of how these metrics can be used for time series prediction and model comparison. The book aims to provide a comprehensive framework for understanding and applying RKHS embeddings in the context of random processes, offering both theoretical insights and practical applications.