Mod ϕ Convergence : Normality Zones and Precise Deviations The canonical way to establish the central limit theorem for i.i.d. random variables is to use characteristic functions and Lévy’s continuity theorem. This monograph focuses on this characteristic function approach and presents a renormalization theory called mod ϕ convergence. This type of convergence is a relatively new concept with many deep ramifications and has not previously been published in a single accessible volume. The authors construct an extremely flexible framework using this concept in order to study limit theorems and large deviations for a number of probabilistic models related to classical probability combinatorics non commutative random variables as well as geometric and number theoretical objects. Intended for researchers in probability theory the text is carefully well written and well structured containing a great amount of detail and interesting examples. . Preface Introduction Preliminaries Fluctuations in the case of lattice distributions Fluctuations in the non lattice case An extended deviation result from bounds on cumulants A precise version of the Ellis Gärtner theorem Examples with an explicit generating function Mod Gaussian convergence from a factorisation of the PGF Dependency graphs and mod Gaussian convergence Subgraph count statistics in Erdös Rényi random graphs Random character values from central measures on partitions Bibliography.