Convergence of Dependent Random Variables : Central Limit Theorems, Berry-Esseen Bounds, Martingale-like Sequences, C-sequences, Strong Laws

Central Limit Theorems, Rates of Convergence are derived for dependent random variables, with relaxed conditions on the dependence. Most of known mixing conditions like strong (alpha-) mixing, absolute regular (beta-mixing),... will satisfy them. This new notion of measure of dependence is developed naturally from the classical Characteristic Function Method, less intuitive but may be more suitable in applications than mixing ones. As it is born from the well-known tool for independent r.v.s's Limit Theorems. Theorems and examples given here prove this notion. Otherwise, it may reach the limit in process of defining measure of the dependence, as argued in this book. On the other aspect, almost sure convergence of adapted sequence, especially of Martingale-like one, is discussed. C-sequence is created, showed not comparative with Amart, Martingale-in-the-limit, by examples. It also is a natural extension of Martingale, derived by seeking condition ensuring a.s. convergence. Also, a phi-mixing Strong Law and some examples of Linear Process are given.