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Mod-\(\Phi\) convergence. Normality zones and precise deviations. (English) Zbl 1387.60003
SpringerBriefs in Probability and Mathematical Statistics. Cham: Springer (ISBN 978-3-319-46821-1/pbk; 978-3-319-46822-8/ebook). xii, 152 p. (2016).
This book deals with different aspects of mod-\(\phi\) convergence to prove precise large or moderate deviations for quite general sequences of real-valued random variables \((X_n)_{n\in\mathbb{N}}\), which can be lattice or non-lattice distributed. Their precise estimates of the fluctuations \(P\{X_n\in tB_n\}\) are established, instead of usual estimates for the rate of exponential decay of \(\log(P\{X_n\in tB_n\})\). The characteristic function approach, for which a renormalization theory called mod-\(\phi\) convergence is proposed.
The approach provides a systematic way to characterize the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. The book is divided into 11 chapters.
The first chapters are devoted to a proof of these abstract results and comparisons with existing results. The new examples covered by this theory come from various areas of mathematics such as classical probability theory, number theory, combinatorics (statistics of additive arithmetic functions), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erdős-Rényi graphs), and many other examples of weakly dependent structures.
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. The large number and the variety of examples hint at a university class for second-order fluctuations.
In summary, this is a useful book for a researcher in probability theory and mathematical statistics. It is very carefully written and collects mane new results.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60E10 Characteristic functions; other transforms
60E05 Probability distributions: general theory
60F10 Large deviations
60F05 Central limit and other weak theorems
60C05 Combinatorial probability
60B20 Random matrices (probabilistic aspects)
05C80 Random graphs (graph-theoretic aspects)
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