SpringerBriefs in Probability and Mathematical Statistics

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1 SpringerBriefs in Probability and Mathematical Statistics Editor-in-chief Mark Podolskij, Aarhus C, Denmark Series editors Nina Gantert, Münster, Germany Richard Nickl, Cambridge, UK Sandrine Péché, Paris, France Gesine Reinert, Oxford, UK Mathieu Rosenbaum, Paris, France Wei Biao Wu, Chicago, USA More information about this series at

2 Valentin Féray Pierre-Loïc Méliot Ashkan Nikeghbali Mod-φ Convergence Normality Zones and Precise Deviations 123

3 Valentin Féray Institut für Mathematik Universität Zürich Winterthurerstrasse Zürich, Switzerland Ashkan Nikeghbali Institut für Mathematik Universität Zürich Winterthurerstrasse Zürich, Switzerland Pierre-Loïc Méliot Laboratoire de Mathématiques Bâtiment 425 Faculté Des Sciences d Orsay Université Paris-Sud Orsay, France ISSN ISSN (electronic) SpringerBriefs in Probability and Mathematical Statistics ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification (2010): 60F05, 60F10, 60C05, 60B20, 05C80 The Author(s) 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

4 ABSTRACT. In this book, we use the framework of mod-φ convergence to prove precise large or moderate deviations for quite general sequences of real-valued random variables (X n ) n N, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations P[X n t n B], instead of the usual estimates for the rate of exponential decay log(p[x n t n B]). Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first chapters of the book are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics. In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of weakly dependent random variables. The large number and the variety of examples hint at a universality class for second-order fluctuations. v

5 Preface The central limit theorem (see [dm56, Lap40, Lya00, Lya01, Lin22, Lév25]) for independent and identically distributed (i.i.d.) random variables is one of the most ubiquitous theorems in probability theory: suitably renormalised, the sum of such variables converges towards a standard normal variable. We refer to [Fis11] for an historical account of this classical result. Roughly speaking, it shows the universality of fluctuations around the first-order limit given by the law of large numbers. Although independence is often used as a first approximation, many natural phenomena exhibit some complex dependency structure. Therefore, a large body of literature is devoted to relaxing the independence hypothesis. Thus, central limit theorems have been given for random variables with an underlying structure such as martingales ([HH80, Chapter3]), Markovchains (see,e.g. [Cog72, IL71, GL78], and [Jon04] for a survey), mixing sequences (see [Ros56, Phi69, Dav73, LL96], and [Bra05] for a survey on mixing conditions), lattice models ([GJL75, EN78, New80]), m-dependence (cf. [RH48, Dia55, Ber73]), dependency graphs ([PL83, Jan88, BR89, Mik91]), exchangeable pairs ([BC05, CGS11,Ros11])and determinantal point processes (cf. [HKPV09, Section 4.6]). vii

6 viii PREFACE Concretely, the central limit theorem for X n expresses the limit of the tail probability when t isafixedrealnumber: lim P[X n E[X n ] tσ n ]= 1 + e s2 2 ds. (1) n 2π Such estimates are crucial in statistics to build confidence intervals. The convergence in law towards a Gaussian distribution is often complemented by other asymptotic results: the speed of convergence (uniformly in t) in Equation (1) (see [Ber41] and [Fel71, Chapter XVI]); the behaviour of the left-hand side of Eq. (1) whent tends to infinity together with n (moderate and large deviation results, see [DZ98]); a subsequent question is how fast can t grow,sothatthe limit given in Eq. (1) is still valid (normality zone); in another direction, local limit theorems describe the probability of X n E[X n ] to be in an interval of constant scale (see [Gne48]). 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 is also used in some of the above-mentioned extensions, in addition to other techniques, mainly based on the method of moments and Stein s method (see for instance the first chapters of [Str11]). t In this monograph, we focus on the characteristic function approach, forwhichweproposearenormalisation theory called mod-φ convergence. If the characteristic function converges after a suitable renormalisation, we prove some precise moderate and large deviation results, which enables us to describe the normality zone. Results for the speed of convergence and local limit theorems will be discussed in a companion work [FMN16]. The idea of using estimates on characteristic functions to obtain central limit theorems and deviation probabilities is of course not new, but we provide here a general framework, together with many examples. These examples come from various mathematical fields: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie

7 ACKNOWLEDGEMENTS ix groups), graph theory (number of subgraphs in a random Erdős- Rényi graph) and non-commutative probability theory (asymptotics of random character values of symmetric groups). Acknowledgements The authors would like to thank Martin Wahl for his input at the beginning of this project and for sharing with us his ideas. We would also like to address special thanks to Andrew Barbour, Reda Chhaibi and Kenny Maples for many fruitful discussions which helped us improve some of our arguments. Zürich, Switzerland Orsay, France Zürich, Switzerland Valentin Féray Pierre-Loïc Méliot Ashkan Nikeghbali

8 Contents Preface Acknowledgements vii ix Chapter 1. Introduction Mod-φ convergence Theoretical results Applications Forthcoming works Discussion on our hypotheses 7 Chapter 2. Preliminaries Basic examples of mod-convergence Legendre-Fenchel transforms Gaussian integrals 14 Chapter 3. Fluctuations in the case of lattice distributions Lattice and non-lattice distributions Deviations at the scale O(t n ) Central limit theorem at the scales o(t n ) and o((t n ) 2/3 ) 26 Chapter 4. Fluctuations in the non-lattice case Berry-Esseen estimates Deviations at scale O(t n ) Central limit theorem at the scales o(t n ) and o((t n ) 2/3 ) Normality zones for mod-φ and mod-gaussian sequences Discussion and refinements 48 Chapter 5. An extended deviation result from bounds on cumulants Bounds on cumulants and mod-gaussian convergence Precise deviations for random variables with control on cumulants Link with the Cramér-Petrov expansion 57 Chapter 6. A precise version of the Ellis-Gärtner theorem Technical preliminaries A precise upper bound A precise lower bound 62 xi

9 xii CONTENTS Chapter 7. Examples with an explicit generating function Mod-convergence from an explicit formula for the Laplace transform Additive arithmetic functions of random integers Number of cycles in weighted probability measure Rises in random permutations Characteristic polynomials of random matrices in a compact Lie group 82 Chapter 8. Mod-Gaussian convergence from a factorisation of the PGF Background: central limit theorem from location of zeros Mod-Gaussian convergence from non-negative factorisations Two examples: uniform permutations and uniform set-partitions Third cumulant of the number of blocks in uniform set-partitions 93 Chapter 9. Dependency graphs and mod-gaussian convergence The theory of dependency graphs Joint cumulants Useful combinatorial lemmas Proof of the bound on cumulants Sums of random variables with a sparse dependency graph 109 Chapter 10. Subgraph count statistics in Erdős-Rényi random graphs A bound on cumulants Polynomiality of cumulants Moderate deviations for subgraph count statistics 117 Chapter 11. Random character values from central measures on partitions Background Definitions and results Technical preliminaries Bounds and limits of the cumulants Asymptotics of the random character values and partitions 136 Bibliography 141