This book offers a comprehensive exploration of probabilistic results, focusing on convergence of probability measures, central limit theorems for sums of random vectors, and invariance principles. It delves into stationary random processes and fields on lattices, slowly varying functions, and various probabilistic inequalities, including uniformly integrable sequences. The text examines weak dependence conditions for random processes and fields, providing classical examples and specific cases such as Davydov’s and Herrndorf’s examples, as well as Gaussian random sequences. It further investigates the asymptotic behavior of variance and estimates on moments for sums of weakly dependent random variables, along with relevant probabilistic inequalities. Methodologies such as Bernstein’s, Gordin’s, and Stein’s methods are discussed, leading to limit theorems for random processes, particularly those that are ?-mixing. The book also covers generalized mixing conditions and their implications for limit theorems, including those related to non-commutative probability theory. Additionally, it addresses limit theorems for random fields, the description of random fields through conditional probability, and the existence and uniqueness of Gibbs random fields. The work concludes with insights into the thermodynamical limit, free energy, and cluster properties, supplemented by references for further study.
