The canonical way to establish the central limit theorem for i.i.d. random variables is to use characteristic functions and L?vys continuity theorem. This monograph focuses on this characteristic function approach and presents a renormalization theory called mod-? 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-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. The book is well written and mathematically rigorous. They authors collect a large variety of results and try to parallel the theory with applications and they do this rather successfully. It may become a standard reference for researchers working on the topic of central limit theorems and large deviation. & this is a useful book for a researcher in probability theory and mathematical statistics.lC$
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