Mod-ϕ Convergence

Normality Zones and Precise Deviations

Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali (Autoren)

Buch | Softcover

XII, 152 Seiten

2016 | 1st ed. 2016
Springer International Publishing (Verlag)
978-3-319-46821-1 (ISBN)

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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- 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.

"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)

Erscheinungsdatum	06.01.2017
Reihe/Serie	SpringerBriefs in Probability and Mathematical Statistics
Zusatzinfo	XII, 152 p. 17 illus., 9 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	Algebra • combinatorics • Combinatorics and graph theory • Deviations • Linear and Multilinear Algebras, Matrix Theory • mathematics and statistics • matrix theory • Number Theory • probability and statistics • Probability Theory • Probability theory and stochastic processes • stochastics
ISBN-10	3-319-46821-9 / 3319468219
ISBN-13	978-3-319-46821-1 / 9783319468211
Zustand	Neuware