This thesis presents a study on metrics between stochastic processes using the method of embedding probability distributions into a Hilbert space with a reproducing kernel. The author, Edgar Alirio Valencia Angulo, is a researcher and professor at the Universidad Tecnológica de Pereira, with expertise in mathematics and probability theory. The work explores the use of reproducing kernel Hilbert spaces (RKHS) to develop metrics for comparing probability distributions and stochastic processes. Key contributions include the development of analytical distance metrics between probability distributions using RKHS, the application of these metrics to hidden Markov models (HMMs), and the estimation of autoregressive processes in RKHS. The study also addresses the challenges of applying these methods in non-i.i.d. settings, such as in HMMs and autoregressive models. The author provides theoretical foundations, mathematical proofs, and practical examples, including the use of Gaussian and Laplacian kernels. The work concludes with experimental results and applications in time series prediction and classification. The thesis is structured into four chapters, covering the theoretical background of RKHS, metrics between probability distributions, metrics between stochastic processes, and applications in time series analysis. The research contributes to the field of machine learning and statistical analysis by offering new tools for comparing and modeling complex stochastic processes.This thesis presents a study on metrics between stochastic processes using the method of embedding probability distributions into a Hilbert space with a reproducing kernel. The author, Edgar Alirio Valencia Angulo, is a researcher and professor at the Universidad Tecnológica de Pereira, with expertise in mathematics and probability theory. The work explores the use of reproducing kernel Hilbert spaces (RKHS) to develop metrics for comparing probability distributions and stochastic processes. Key contributions include the development of analytical distance metrics between probability distributions using RKHS, the application of these metrics to hidden Markov models (HMMs), and the estimation of autoregressive processes in RKHS. The study also addresses the challenges of applying these methods in non-i.i.d. settings, such as in HMMs and autoregressive models. The author provides theoretical foundations, mathematical proofs, and practical examples, including the use of Gaussian and Laplacian kernels. The work concludes with experimental results and applications in time series prediction and classification. The thesis is structured into four chapters, covering the theoretical background of RKHS, metrics between probability distributions, metrics between stochastic processes, and applications in time series analysis. The research contributes to the field of machine learning and statistical analysis by offering new tools for comparing and modeling complex stochastic processes.