Mod-ϕ Convergence

1. Front Matter

   Pages i-xii

   PDF

2. Introduction

   • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

   Pages 1-8

3. Preliminaries

   • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

   Pages 9-16

4. Fluctuations in the case of lattice distributions

   • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

   Pages 17-32

5. Fluctuations in the non-lattice case

   • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

   Pages 33-50

6. An extended deviation result from bounds on cumulants

   • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

   Pages 51-58

7. A precise version of the Ellis-Gärtner theorem

   • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

   Pages 59-64

8. Examples with an explicit generating function

   • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

   Pages 65-86

9. Mod-Gaussian convergence from a factorisation of the probability generating function

   • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

   Pages 87-94

10. Dependency graphs and mod-Gaussian convergence

    • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

    Pages 95-110

11. Subgraph count statistics in Erdős-Rényi random graphs

    • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

    Pages 111-122

12. Random character values from central measures on partitions

    • Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

    Pages 123-139

13. Back Matter

    Pages 141-152

    PDF

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. 

• Probability Theory
• Number Theory
• Combinatorics
• Matrix Theory
• Deviations

“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. It is very carefully written and collects many new results.” (Nikolai N. Leonenko, zbMATH 1387.60003, 2018)

“This beautiful book (together with other publications by these authors) opens a new way of proving limit theorems in probability theory and related areas such as probabilistic number theory, combinatorics, and statistical mechanics. It will be useful to researchers in these and many other areas.” (Zakhar Kabluchko, Mathematical Reviews, September, 2017)

• Institut für Mathematik, Universität Zürich — Winterthurerstrasse, Zürich, Switzerland

  Valentin Féray, Ashkan Nikeghbali

• Laboratoire de Mathématiques, Bâtiment 425 — Faculté Des Sciences, d’Orsay—Université Paris-Sud, Orsay, France

  Pierre-Loïc Méliot

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