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Manuscritmanuscrit MANUSCRITMANUSCRIT Présenté pour l’obtention de L’HABILITATION À DIRIGER DES RECHERCHES Délivrée par : l’Université Toulouse 3 Paul Sabatier Présentée et soutenue le 10/05/2016 par : Florian SIMATOS Théorèmes limite fonctionnels, processus de branchement et réseaux stochastiques Functional limit theorems, branching processes and stochastic networks JURY Patrick CATTIAUX Professeur des universités Président du jury Jean-François DELMAS Professeur des universités Membre du jury Thomas DUQUESNE Professeur des universités Rapporteur Sergey FOSS Full professor Membre du jury David GAMARNIK Full professor Rapporteur Laurent MICLO Directeur de recherche Membre du jury École doctorale et spécialité : MITT : Domaine Mathématiques : Mathématiques appliquées Unité de Recherche : ISAE-SUPAERO Directeur de Thèse : Laurent MICLO Rapporteurs : Maury BRAMSON , Thomas DUQUESNE et David GAMARNIK i This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. ii Acknowledgements First and foremost, I would like to thank Professor Maury Bramson, Professor Thomas Duquesne and Professor David Gamarnik for accepting to review my manuscript. Some of their papers rank among the most influential in the fields of branching pro- cesses and stochastic networks, and it is an honor to have them as reviewers. I would also like to thank Professor Laurent Miclo for accepting to act as direc- tor of my habilitation, as well as Professor Patrick Cattiaux, Professor Jean-François Delmas and Professor Sergey Foss for being part of my defense committee. I am learning on a daily basis how scarce a resource time is and it makes me even more thankful for the time everybody has devoted to my project. Research is a weird business: it is done 90% alone, but these are the 10% re- maining, made of lively discussions and interactions, which make it so exciting. It is therefore my sincere pleasure to thank here my various co-authors as well some colleagues who have played an important role in my career at some point even though it did not translate into a publication: the work presented here was only possible thanks to them. My warmest thanks thus go to Elie Aïdekon, François Bac- celli, Vincent Bansaye, Sem Borst, Niek Bouman, Ed Coffman, A. Ganesh, Fabrice Guillemin, Tom Kurtz, Peter Lakner, Amaury Lambert, Marc Lelarge, Lasse Leskëla, Sarah Lilienthal, D. Manjunath, Alexandre Proutière, Josh Reed, Philippe Robert, Emmanuel Schertzer, Shuzo Tarumi, Danielle Tibi, Johan van Leeuwaarden, Remco van der Hofstad, Gil Zussman and Bert Zwart. I hope to have the chance – and, more importantly, the time – to pursue our rewarding collaborations. The four-year period that I have spent in the Netherlands between CWI and TU/e represents the most exciting time I have had from a scientific standpoint. Although the previous list only contains a few Dutch names, I would also like to address a general thank to my formeer Dutch colleagues who have made these four years so enjoyable and productive: most of the work presented here has indeed been carried out while in the Netherlands. I would like to thank more specifically Bert Zwart, Sem Borst and Johan van Leeuwaarden for funding my post-doctoral positions. My special thanks to Odile Riteau, our enthusiastic secretary at ISAE-SUPAERO, for her precious help in arranging all the practical details of the defense. Finally, the writing of this manuscript has come at a turning point of my career and personal life: I was hired at ISAE-SUPAERO three months earlier and Gaëtan was barely two-month old. My habilitation and lecture notes being now written, I do not anticipate any more acknowledgements in the near future and so I seize this opportunity to thank one more time Sandrine for her love and support – may it last forever. iii iv Contents Acknowledgements iii Summary vii 1 Introduction 1 1.1 Functional law of large numbers and central limit theorem . .1 1.2 Motivation . .3 1.3 Notation used throughout the document . .6 2 Two theoretical results on weak convergence 11 2.1 Introduction . 11 2.2 A sufficient condition for tightness . 14 2.3 Weak convergence of regenerative processes . 17 3 Branching processes 23 3.1 Introduction . 24 3.2 Galton–Watson processes in varying environments . 30 3.3 Binary, homogeneous Crump–Mode–Jagers processes . 37 3.4 Crump–Mode–Jagers trees with short edges . 45 4 Stochastic networks 53 4.1 Introduction . 53 4.2 Processor-Sharing queue length process . 56 4.3 A stochastic network with mobile customers . 64 4.4 Lingering effect for queue-based scheduling algorithms . 72 5 Branching processes in mathematical finance 81 5.1 Study of a model of limit order book . 81 6 Research perspectives 91 6.1 Branching processes . 91 6.2 Stochastic averaging via functional analysis . 93 6.3 Stochastic networks . 94 v vi CONTENTS Summary Overview This manuscript describes some of the work I have been doing since 2010 and the end of my PhD. As the title suggests, it contains three main parts. Functional limit theorems: Chapter 2 presents two theoretical results on the weak convergence of stochastic processes: one – Theorems 2.2.1 and 2.2.2 – is a suf- ficient condition for the tightness of a sequence of stochastic processes and the other – Theorem 2.3.1 – provides a sufficient condition for the weak convergence of a sequence of regenerative processes; Branching processes: in Chapter 3, scaling limits of three particular types of branch- ing processes are discussed: 1) Galton–Watson processes in varying environments, 2) binary and homogeneous Crump–Mode–Jagers processes and 3) Crump–Mode– Jagers processes with short edges; Stochastic networks: Chapter 4 presents three results on stochastic networks: 1) scaling limits of the M/G/1 Processor-Sharing queue length process, 2) study of a model of stochastic network with mobile customers and 3) heavy traffic delay performance of queue-based scheduling algorithms. These topics are closely intertwined: tightness of Galton–Watson processes in varying environments studied in Section 3.2 is obtained thanks to Theorem 2.2.1 of Chapter 2; results of Section 4.2 on the Processor-Sharing queue are essentially ob- tained by combining results on binary and homogeneous Crump–Mode–Jagers pro- cesses from Section 3.3 together with the sufficient condition for the convergence of regenerative processes from Chapter 2; and scaling limits of the model of mo- bile network of Section 4.3 rely on the same theoretical result. In order to give an overview of how these results relate to each other, Chapters 2, 3 and 4 are briefly presented next. Chapter 5 is devoted to the study of a model of limit order books in mathematical finance: the main technical tool is a coupling with a branching random walk, and it also adopts an excursion point-of-view similar in spirit as in Section 2.3 whereby the limiting process is characterized by its excursion measure. Finally, the manuscript is concluded in Chapter 6 with research perspectives. vii viii CONTENTS Summary of Chapter 2 Section 2.2 presents the main result of [BS, BKS], namely a sufficient condition for the tightness of a sequence of stochastic processes (Xn) of the form h i E 1 d(X (t), X (t))¯ X (u),u s F (t) F (s), n 1,0 s t, ^ n n j n · · n ¡ n ¸ · · J with d the distance, ¯ 0 and (F ) a sequence of càdlàg functions with F 1 F in È n n! the Skorohod J1 topology. This result extends a result from Kurtz [74] where F was assumed to be continuous: allowing general càdlàg F was motivated by the study of processes in varying environments that may exhibit accumulation of fixed points of discontinuity. This criterion is used in Section 3.2 to prove the tightness of a se- quence of Galton–Watson processes in varying environments. Section 2.3 presents the main results of [LS14]. This paper establishes a new method for proving the weak convergence of a sequence of regenerative processes from the convergence of their excursions. It is shown that for a tight sequence (Xn) of regen- erative processes to converge, it is enough that the first excursion with “size” " È converges together with its left- and right-endpoints and its size. Here the size of an excursion is given by a general measurable mapping ' with values in [0, ]. This 1 approach was motivated by the study of the Processor-Sharing queue in Section 4.2 and also proved useful to study a stochastic network with mobile users discussed in Section 4.3. Summary of Chapter 3 Section 3.2 presents results on scaling limits of Galton–Watson processes in vary- ing environments obtained in [BS15]. These results adopt the original approach of Grimvall [44] via the study of Laplace exponents and extend results of Kurtz [75] and Borovkov [17] by allowing offspring distributions to have infinite variance. The price to pay is a loss of generality in the drift term, corresponding to the first moments of the offspring distributions, which is assumed to have finite variation. Section 3.3 presents results on scaling limits of binary and homogeneous Crump– Mode–Jagers processes obtained in [LSZ13, LS15]. The main tool is the connec- tion with Lévy processes, whereby a binary and homogeneous Crump–Mode–Jagers process is seen as the local time process of a compound Poisson process with drift stopped upon hitting 0. Assuming that the sequence of Lévy processes converges, it is shown that this implies weak convergence of the sequence of associated local time processes – and thus of the binary and homogeneous Crump–Mode–Jagers processes – in the finite variance case where the limiting Lévy process is a Brow- nian motion.
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