Synopses & Reviews
Synopsis
The canonical way to establish the central limit theorem for i.i.d. random variables is to use characteristic functions and L vy'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.
Synopsis
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-G rtner 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 Erd s-R nyi random graphs.- Random character values from central measures on partitions.- Bibliography.