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International audience; We focus on the estimation of the intensity of a Poisson process in the presence of a uniform noise. We propose a kernel-based procedure fully calibrated in theory and practice. We show that our adaptive estimator is optimal from the oracle and minimax points of view, and provide new lower bounds when the intensity belongs to a Sobolev ball. By developing the Goldenshluger-Lepski methodology in the case of deconvolution for Pois-son processes, we propose an optimal data-driven selection of the kernel's bandwidth, and we provide a heuristic framework to calibrate the estimator in practice. Our method is illustrated on the spatial repartition of replication origins along the human genome.

This thesis aims at evaluating the size of the Delaunay triangulation of points drawn on a surface with a random distribution. The Delaunay triangulation, is a geometrical object that appeared recurrently in the scientific history. In dimension 2, the Delaunay triangulation is the set of triangles for which the circumscribing circle does not contain other points of X. This definition is generalizable in higher dimensions.Today, the Delaunay triangulation is one the most studied structures in computational geometry. For the 2 dimensional case, we know that the size of the Delaunay triangulation remains linear with the number of points. In 3 dimension, it is not anymore the case. The size of the 3D-Delaunay triangulation can range from linear to quadratic. This size depends on how the points are distributed in R^3. On a surface, the size of the Delaunay triangulation will depend both on the surface and on how they are distributed on this surface.To model points, we choose to use a Poisson point process since it verifies properties of homogeneity and independence that are convenient for the computations. In order to prove the expected O(n log n) bound for the uniform sample distributed on a cylinder, Devillers et al. remarked that the intersection of the cylinder with a sphere passing though two points p and q on the cylinder always contains a specific triangle drawn on the cylinder. That leads them to study a 2-dimensional graph in which two points are neighbors if there exists such a triangle that does not contain other data points. Such a graph has expected size O(n log n), and this is how they obtain the O(n log n) bound. In Part II, we define a kind of empty region graphs, we formalize a method to compute lower and upper bounds on their expected size, and give tight results for such graphs.As Attali et al. pointed out, the intersection of a sphere with a generic surface has almost an elliptic shape, aligned with the curvature directions of the surface. This leads us to study a particular empty region graph for which the regions are axis-aligned ellipses. We prove, in Part II, that if theinvolved ellipses have an aspect ratio ranging from b to 1, with 0 < b < 1, then the expected number of neighbors of any point in the graph is O(ln b).In order to illustrate the method, we compute, in Part III, tight asymptotic bounds on the expected size of the 3D-Delaunay triangulation in two specific cases. In Part III, Chapter 12, we consider a cylinder of revolution and reprove the O(N ln N) bound but for a Poisson point process. Considering the similarity between uniform and Poisson sample, the goal of this chapteris mainly to present concretely the method in a 3-dimensional simple case. Then, in Chapter 13, we compute the size of the 3D-Delaunay triangulation of a Poisson process distributed on a flattened sphere. We show that the expected size of the triangulation is O(N).Finally in Part IV, we treat the case of generic surfaces. Even if an oblate spheroid is a specific surface, we will be able to reuse some computations in this part up to some adaptations. Indeed the oblate spheroid is the surface of a convex body, that is not generally the case. It has a lot of symmetries,that is not generally the case either. In this part, we focus more on how to deal with these adaptations than on the computations that were already quite tedious in the spheroid case., Cette thèse a pour but d' évaluer la taille de la triangulation de Delaunay de points distribués sur une surface avec une distribution aléatoire. La triangulation de Delaunay, est un objet géométrique qui est apparu de manière récurrente dans l'histoire scientifique. En dimension 2, la triangulation de Delaunay de X est l'ensemble des triangles pour lesquels le cercle circonscrit ne contient pas d'autres points de X. Cette définition est généralisable en dimension supérieure.Aujourd'hui, la triangulation de Delaunay est l'une des structures les plus étudiées en géométrie algorithmique. Pour le cas en 2 dimensions, on sait que la taille de la triangulation de Delaunay reste linéaire avec le nombre de points. En 3 dimensions, la taille de la triangulation de Delaunay peut varier de linéaire à quadratique. Cette taille dépend de la façon dont les points sont distribués dans R^3. Sur une surface, la taille de la triangulation de Delaunay dépendra à la fois de la surface et de la façon dont ils sont répartis sur cette surface.Pour modéliser les points, nous choisissons d'utiliser un processus ponctuel de Poisson car il vérifie des propriétés d'homogénéité et d'indépendance qui sont pratiques pour les calculs. Afin de prouver la borne O(n log n) en espérance pour l'échantillon uniforme distribué sur un cylindre, Devillers et al. ont remarqué que l'intersection du cylindre avec une sphère passant par deux points p et q sur le cylindre contient toujours un triangle spécifique dessiné sur le cylindre. Cela les conduit à étudier un graphe à 2 dimensions dans lequel deux points sont voisins s'il existe un tel triangle qui ne contient pas d'autres points de données. Un tel graphe a une taille moyenne de O(n log n), et c'est ainsi qu'ils obtiennent la borne O(n log n). Dans la partie II, nous définissons un type de graphes de régions vides, nous formalisons une méthode pour calculer les limites inférieures et supérieures de leur taille moyenne, et nous donnons des résultats optimaux pour ces graphes.Comme Attali et al. l'ont souligné, l'intersection d'une sphère avec une surface générique a presque une forme elliptique, alignée avec les directions de courbure de la surface. Ceci nous amène à étudier un graphe particulier de régions vides pour lequel les régions sont des ellipses alignées avec les axes. Nous prouvons, dans la partie II, que si les ellipses concernées ont un rapport d'aspect compris entre b et 1, avec 0 < b < 1, alors le nombre moyen de voisins de tout point du graphe est O(ln b).Afin d'illustrer la méthode, nous calculons, dans la partie III, les limites asymptotiques sur la taille moyenne de la triangulation 3D-Delaunay dans deux cas spécifiques. Dans la partie III, chapitre 12, nous considérons un cylindre de révolution et confirmons la limite O(n ln n) mais pour un processus ponctuel de Poisson. Compte tenu de la similitude entre l'échantillon uniforme et l'échantillon de Poisson, le but de ce chapitreest principalement de présenter concrètement la méthode dans un cas simple en 3 dimensions. Ensuite, dans le chapitre 13, nous calculons la taille de la triangulation 3D-Delaunay d'un processus de Poisson distribué sur une sphère aplatie. Nous montrons que la taille moyenne de la triangulation est O(n).Enfin, dans la partie IV, nous traitons le cas des surfaces génériques. Même si un sphéroïde aplati est une surface spécifique, nous pourrons réutiliser certains calculs de cette partie moyennant quelques adaptations. En effet, le sphéroïde aplati est la surface d'un corps convexe, ce qui n'est généralement pas le cas en général. Dans cette partie, nous nous concentrons plus sur la façon de gérer ces adaptations que sur les calculs qui étaient déjà assez fastidieux dans le cas du sphéroïde.

In this note, we propose a slightly different proof of Gallavotti's theorem ["Statistical Mechanics: A Short Treatise", Springer, 1999, pp. 48--55] on the derivation of the linear Boltzmann equation for the Lorentz gas with a Poisson distribution of obstacles in the Boltzmann-Grad limit., Comment: 18 pages

Beam management is central in the operation of beamformed wireless cellular systems such as 5G New Radio (NR) networks. Focusing the energy radiated to mobile terminals (MTs) by increasing the number of beams per cell increases signal power and decreases interference, and has hence the potential to bring major improvements on area spectral efficiency (ASE). This paper proposes a first system-level stochastic geometry model encompassing major aspects of the beam management problem: frequencies, antenna configurations, and propagation; physical layer, wireless links, and coding; network geometry, interference, and resource sharing; sensing, signaling, and mobility management. This model leads to a simple analytical expression for the effective rate that the typical user gets in this context. This in turn allows one to find the number of beams per cell and per MT that maximizes the effective ASE by offering the best tradeoff between beamforming gains and beam management operational overheads and costs, for a wide variety of 5G network scenarios including millimeter wave (mmWave) and sub-6 GHz. As part of the system-level analysis, we define and analyze several underlying new and fundamental performance metrics that are of independent interest. The numerical results discuss the effects of different systemic tradeoffs and performance optimizations of mmWave and sub-6 GHz 5G deployments., Comment: This is an extended version of the paper that was accepted at IEEE GLOBECOM 2020. Comments are welcome. arXiv admin note: text overlap with arXiv:2006.05027

International audience; Wire and Arc Additive Manufacturing (WAAM) is a direct-energy deposition technique (unlike SLM or EBM) that builds up a part in a layer-by-layer fashion, each layer being constituted of interlaced weld beads. It is the best suited Additive Manufacturing (AM) technique for large structures thanks to its high deposition rate (5 kg/h). The resulting material shows a rough surface, strong residual stress induced by its complex thermal history, a heterogeneous microstructure marked by the different weld passes as well as defects formed by gas pockets. Despite their rarity, pores are found to have a first-order influence on the fatigue life of machined specimens. The discrepancy in their size (> 100 μm) and position is responsible for a considerable scatter that makes classical fatigue tests ineffective. The aim of this study is to propose a novel approach to take into account the effect of rare WAAM-induced defects in high cycle fatigue. To achieve this, numerical porous structures are generated from the knowledge of the real pore population determined by tomography. Their fatigue performances are predicted via a two-scale probabilistic model identified on experimental self-heating results, on which pores have no influence. In that sense, the probabilistic model describes the behavior of a virtually healthy material. Then, by computing a database of representative pore cases, the whole bundle of Wöhler curves for each numerical porous structure is determined. Finally, the numerical fatigue scatter is in close agreement with experimental data, and it is shown that the ranking in pore criticality according to the model matches the fractography observations.

We define a random model for the moments of the new eigenfunctions of a point scatterer on a two-dimensional rectangular flat torus. In the deterministic setting, Seba conjectured these moments to be asymptotically Gaussian, in the semi-classical limit. This conjecture was disproved by Kurlberg–Ueberschar on Diophantine tori. In our model, we describe the accumulation points in distribution of the randomized moments, in the semi-classical limit. We prove that asymptotic Gaussianity holds if and only ifs some function, modeling the multiplicities of the Laplace eigenfunctions, diverges to $$+\infty $$ .

We study a class of reinforcement models involving a Poisson process on the vertices of certain infinite graphs G. When a vertex fires, one of the edges incident to that vertex is selected. The edge selection is biased towards edges that have been selected many times previously, and a parameter α governs the strength of this bias. We show that for various graphs (including all graphs of bounded degree), if α 1 (the very strong reinforcement regime) then the random subgraph consisting of edges that are ever selected by this process does not percolate (all connected components are finite). Combined with results appearing in a companion paper, this proves that on these graphs, with α sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of nonpercolation relies on coupling with a percolation-type model that may be of interest in its own right.

International audience; Inspired by the subject of performance enhancement of users located at the edge of cells in cellular networks, this paper studies a centralized frequency reuse scheme depending on the link quality of users within the Voronoi cell of the serving base station (BS) for the homogeneous Poisson point process (PPP) network. To quantify how much a user experiencing performance degradation, we consider that each cell is partitioned into multiple regions based on the strength of the signal to interference ratio. In our model, a given resource block (RB) is allocated to one user in a cell and cannot be shared by another user in this cell. The RB is divided into multiple sub-channels and the user of interest uses only one of the multiple sub-channels, the mapping depending on the region in which the user belongs, letting a part of the RB unused to reduce the interference to the other cells. This scenario implies that the interfering set of BS depends on the coverage probability of the typical user. We tightly approximate the interfering BS set as a thinned version of the original PPP depending on the probability of coverage of each type of user. Results show that the scheme increases the network's global coverage probability and enhances spectral efficiency by comparing the known fractional frequency reuse scheme.

International audience; This paper presents a non-full interference analysis to derive the coverage probability when the shared spectrum allocation technique is applied to the homogeneous Poisson point process (PPP) network. Each cell is partitioned into two regions based on the strength of the signal to interference ratio (SINR): the cell center region if the SINR is more significant than a threshold and cell edge region otherwise. In our model, a given resource block (RB) is allocated to one user in a cell and cannot be shared by another user. The RB is hence divided into two sub-bands, which will be used exclusively according to whether the user is a cell center or cell edge user, letting a part of the RB unused in order to reduce the interference to the other cell. This scenario implies that the interfering set of base stations (BS) depends on the coverage probability of the typical user. We prove that the interfering BS set is a thinned version of the original PPP and is related to the probability of coverage of the cell center region. Results show that the scheme increases the network's global coverage probability.

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is either a conditional Gaussian vector or a conditional Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Monge-Kantorovich transport distance in the univariate case; and for an other probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by Peccati, Solé, Taqqu & Utzet [31] for Gaussian approximation; and by Peccati [32] for Poisson approximations, as well as the recently established fourth-moment theorems on the Poisson space of Döbler & Peccati [7]. Applications to stochastic processes are given.

The main focus of this thesis is on modeling, performance evaluation and system-level optimization of next-generation cellular networks by using stochastic geometry. In addition, the emerging technology of Reconfigurable Intelligent Surfaces (RISs) is investigated for application to future wireless networks. In particular, relying on a Poisson-based abstraction model for the spatial distribution of nodes and access points, this thesis develops a set of new analytical frameworks for the computation of important performance metrics, such as the coverage probability and potential spectral efficiency, which can be used for system-level analysis and optimization. More specifically, a new analytical methodology for the analysis of three-dimensional cellular networks is introduced and employed for system optimization. A novel resource allocation problem is formulated and solved by jointly combining for the first time stochastic geometry and mixed-integer non-linear programming. The impact of deploying intelligent reflecting surfaces throughout a wireless network is quantified with the aid of line point processes, and the potential benefits of RISs against relaying are investigated with the aid of numerical simulations.; Cette thèse porte principalement sur la modélisation, l'évaluation des performances et l'optimisation au niveau système des réseaux cellulaires de nouvelle génération à l'aide de la géométrie stochastique. En plus, la technologie émergente des surfaces intelligentes reconfigurables (RISs) est étudiée pour l'application aux futurs réseaux sans fil. En particulier, reposant sur un modèle d’abstraction basé sur la loi de Poisson pour la distribution spatiale des nœuds et des points d’accès, cette thèse développe un ensemble de nouveaux cadres analytiques pour le calcul d’importantes métriques de performance, telles que la probabilité de couverture et l'efficacité spectrale potentielle, qui peuvent être utilisés pour l'analyse et l'optimisation au niveau système. Plus spécifiquement, une nouvelle méthodologie d'analyse pour l'analyse de réseaux cellulaires tridimensionnels est introduite et utilisée pour l'optimisation du système. Un nouveau problème d’allocation de ressources est formulé et résolu en combinant pour la première fois géométrie stochastique et programmation non linéaire mixte en nombres entiers. L'impact du déploiement de surfaces réfléchissantes intelligentes sur un réseau sans fil est quantifié à l'aide de processus ponctuels, et les avantages potentiels des RISs contre le relais sont étudiés à l'aide de simulations numériques.

The main focus of this thesis is on modeling, performance evaluation and system-level optimization of next-generation cellular networks by using stochastic geometry. In addition, the emerging technology of Reconfigurable Intelligent Surfaces (RISs) is investigated for application to future wireless networks. In particular, relying on a Poisson-based abstraction model for the spatial distribution of nodes and access points, this thesis develops a set of new analytical frameworks for the computation of important performance metrics, such as the coverage probability and potential spectral efficiency, which can be used for system-level analysis and optimization. More specifically, a new analytical methodology for the analysis of three-dimensional cellular networks is introduced and employed for system optimization. A novel resource allocation problem is formulated and solved by jointly combining for the first time stochastic geometry and mixed-integer non-linear programming. The impact of deploying intelligent reflecting surfaces throughout a wireless network is quantified with the aid of line point processes, and the potential benefits of RISs against relaying are investigated with the aid of numerical simulations., Cette thèse porte principalement sur la modélisation, l'évaluation des performances et l'optimisation au niveau système des réseaux cellulaires de nouvelle génération à l'aide de la géométrie stochastique. En plus, la technologie émergente des surfaces intelligentes reconfigurables (RISs) est étudiée pour l'application aux futurs réseaux sans fil. En particulier, reposant sur un modèle d’abstraction basé sur la loi de Poisson pour la distribution spatiale des nœuds et des points d’accès, cette thèse développe un ensemble de nouveaux cadres analytiques pour le calcul d’importantes métriques de performance, telles que la probabilité de couverture et l'efficacité spectrale potentielle, qui peuvent être utilisés pour l'analyse et l'optimisation au niveau système. Plus spécifiquement, une nouvelle méthodologie d'analyse pour l'analyse de réseaux cellulaires tridimensionnels est introduite et utilisée pour l'optimisation du système. Un nouveau problème d’allocation de ressources est formulé et résolu en combinant pour la première fois géométrie stochastique et programmation non linéaire mixte en nombres entiers. L'impact du déploiement de surfaces réfléchissantes intelligentes sur un réseau sans fil est quantifié à l'aide de processus ponctuels, et les avantages potentiels des RISs contre le relais sont étudiés à l'aide de simulations numériques.

Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuum space version of scale-free percolation introduced in [DW18]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in $\mathbb R^d$. Each vertex is equipped with a random weight and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the the annealed clustering coefficient of one point, which is a strictly positive quantity., 20 pages

Summary The increase in availability of species data sets means that approaches to species distribution modelling that incorporate multiple data sets are in greater demand. Recent methodological developments in this area have led to combined likelihood approaches, in which a log-likelihood comprised of the sum of the log-likelihood components of each data source is maximised. Often, these approaches make use of at least one presence-only data set and use the log-likelihood of an inhomogeneous Poisson point process model in the combined likelihood construction. While these advancements have been shown to improve predictive performance, they do not currently address challenges in presence-only modelling such as checking and correcting for violations of the independence assumption of a Poisson point process model or more general challenges in species distribution modelling such as overfitting. In this paper, we present an extension of the combined likelihood framework which accommodates alternative presence-only likelihoods in the presence of spatial dependence as well as lasso-type penalties to account for potential overfitting. We compare the proposed approach combined penalised likelihood approach to the standard combined likelihood approach via simulation and apply the method to modelling the distribution of the Eurasian lynx in the Jura Mountains in eastern France. The simulations show that the proposed combined penalised likelihood approach outperforms the standard approach when spatial dependence is present in the data. The lynx analysis shows that the predicted maps vary significantly with the different implementations of the proposed approach. This work highlights the benefits of careful consideration of the presence-only components of the combined likelihood formulation, and allows greater flexibility and ability to accommodate real datasets.

International audience; Understanding the processes related to wildlife recoveries is not only essential in solving humanwildlife conflicts, but also for identifying priority conservation areas and in turn, for effective conservation planning. We used data from a citizen science program to study spatial aspects of the demographic and genetic recovery of brown bears in Greece and to identify new areas for their conservation. We visually compared our data with an estimation of the past distribution of brown bears in Greece and used a point process approach to model habitat suitability. We then compared our results with the current distribution of brown bear records and with that of protected areas. Our results indicate that in the last 15 years bears may have increased their range by as much as 100%, by occupying mainly anthropogenic landscapes and areas with suitable habitat that are currently not legally protected, thus creating a new conservation reality for the species in Greece. This development dictates the re-evaluation of the national management and conservation priorities for brown bears in Greece by focusing in establishing new protected areas that will safeguard their recovery. Our conservation approach is a swift and cheap way of identifying priority conservation areas, while gaining important insights on spatial aspects of population recovery. It will help prioritize conservation actions for brown bears in Greece and may serve as a model conservation approach to countries facing similar financial and logistic constraints in the monitoring of local biodiversity or facing challenges in managing rapid population recoveries. Our conservation approach appeared to be particularly suited to identifying priority areas for conservation in areas with recovering wildlife populations and may therefore be used as an 'early-warning' conservation system.

International audience; The absorption mechanisms of Poisson-distributed disordered Metal-Insulator-Metal structures are investigated theoretically and numerically. In particular, from the Poisson point process, Slivnyak-Mecke integrals are used to predict the number of isolated particles and majority clusters as a function of the particle density z. Then, the impact of these numbers on peak, mean, and bandwidth absorbance is calculated by full-wave electromagnetic analysis. It is shown that the peak absorbance reaches a maximum value of higher than 0.9 for a particle density of 0.6 cm-2 originating from main contributions of monomers and dimers whereas trimers represent less than 6% of the total number of particles. The conclusions of these studies are experimentally confirmed by means of free-space characterization.

The main result of this paper states that if $(N, \Pi)$ is a pair of independent point processes on a common ground space with $N$ Poisson and $\Pi$ determinantal induced by a locally trace class (not necessarily self-adjoint) correlation kernel, then their independent superposition $N + \Pi$ determines uniquely the distributions of $N$ and $\Pi$., Comment: 7 pages

International audience; For a wide class of stationary time series, extreme value theory provides limiting distributions for rare events. The theory describes not only the size of extremes, but also how often they occur. In practice, it is often observed that extremes cluster in time. Such short-range clustering is also accommodated by extreme value theory via the so-called extremal index. This review provides an introduction to the extremal index by working through a number of its intuitive interpretations. Thus, depending on the context, the extremal index may represent (i) the loss of iid degrees of freedom, (ii) the multiplicity of a compound Poisson point process, (iii) the inverse mean duration of extreme clusters. More recently, the extremal index has also been used to quantify (iv) recurrences around unstable fixed points in dynamical systems. Whether extreme events occur in isolation or in clusters is an important question for their prediction and mitigation. Extreme value theory can accommodate clustering via the extremal index which, heuristically, measures the size of the cluster. Mathematically, clustering is most conveniently understood within the framework of point processes. Without clustering, extreme events occur in the manner of a Poisson process. With clustering, extreme events are bunched together in a compound Poisson process. In order to develop an intuition for the extremal index, we survey some simple examples from stochastic processes , real-world time series and dynamical systems .

A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of $\mathbb{R}^n$ or a stationary Poisson point process in $\mathbb{R}^n$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension $n$ to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and in source coding., 29 pages

Consider a continuous-state branching population constructed as a flow of nested subordinators. Inverting the subordinators and reversing time give rise to a flow of coalescing Markov processes (with negative jumps) which correspond to the ancestral lineages of individuals in the current generation. The process of the ancestral lineage of a fixed individual is the Siegmund dual process of the continuous-state branching process. We study its semi-group, its long-term behavior and its generator. In order to follow the coalescences in the ancestral lineages and to describe the backward genealogy of the population, we define non-exchangeable Markovian coalescent processes obtained by sampling independent Poisson arrival times over the flow. These coalescent processes are called consecutive coalescents, as only consecutive blocks can merge. They are characterized in law by finite measures on $\mathbb{N}$ which can be thought as the offspring distributions of some inhomogeneous immortal Galton-Watson processes forward in time., 56 pages, 6 figures

International audience; This paper introduces a 3-D model for wireless communication systems in order to evaluate their performance under hybrid fading assumption for LOS and non-LOS environments. The model is based on the concepts of 3-D stochastic geometry that describes the different cells positions in the network with realistic constraints, and the random shape theory which characterizes the random blockages in urban cellular networks. Here, we assume that the blockages are randomly distributed as a Boolean scheme of rectangles. Based on that, probabilities of line-of-sight (LOS), for 2-D and 3-D models, are evaluated, and the corresponding coverage probability expressions are detailed and derived under the assumption of Rayleigh fading, Rice fading and hybrid fading. Using numerical results, the analytical derived expressions are evaluated, and the advantages of employing the more realistic 3-D model are investigated.

International audience; A device-to-device (D2D) wireless network is considered, where user devices also have the ability to cache content. In such networks, users are mobile and communication links can be spontaneously activated and dropped depending on the users’ relative position. Receivers request files from transmitters, these files having a certain popularity and file-size distribution. In this work a new performance metric is introduced, namely the Service Success Probability, which captures the specificities of D2D networks for the Poisson Point Process case of node distribution and both the SNR and SINR coverage model. For all scenarios, the success probability depends not just on the content placement policy, but essentially on the transmitter association policy. Association can either be cache-aware or cache-unaware depending on information availability over cache inventories. We derive explicit expressions related to the defined performance metric using stochastic geometry analysis. We further extend the analysis from one-shot transmission to consecutive transmitter associations, where the receiver is allowed to profit from node mobility and gradually receive service by associating stepwise to nodes in its vicinity. Simulations support the analytical results and explain the influence of mobility, association, fading, file-size distribution and frequency reuse on the D2D performance, while providing intuition on how to appropriately cache content on mobile storage space. Of particular interest is the investigation on how different file-size distributions (Exponential, Uniform, or Heavy-Tailed) influence the performance. Finally, it is illustrated that D2D communications considerably benefit from multiple consecutive associations, which can guarantee service completion especially in high mobility environments.

In communication networks, system-level analysis and optimization are useful when one is interested in optimizing the system performance across the entire network. System-level analysis and optimization, therefore, are relevant for optimally operating current networks, and for deploying and planning future networks. In the last few years, the system-level modeling and analysis of cellular networks have been facilitated by capitalizing on the mathematical tool of stochastic geometry and, more precisely, on the theory of spatial point processes. It has been empirically validated that, from the system-level standpoint, the locations of cellular base stations can be abstracted as points of a homogeneous Poisson point process whose intensity coincides with the average number of based stations per unit area.In this context, the contribution of the present Ph.D. thesis lies in developing new analytical methodologies for analyzing and optimizing emerging cellular network deployments. The present Ph.D. thesis, in particular, provides three main contributions to the analysis and optimization of energy-efficient cellular networks.The first contribution consists of introducing a tractable approach for assessing the feasibility of multiple-antenna cellular networks, where low-energy mobile devices decode data and harvest power from the same received signal. Tools from stochastic geometry are used to quantify the information rate vs. harvested power tradeoff. Our study unveils that large-scale antenna arrays and ultra-dense deployments of base stations are both necessary to harvest, with high reliability, a sufficiently high amount of power. Furthermore, the feasibility of receiver diversity for application to downlink cellular networks is investigated. Several options that are based on selection combining and maximum ratio combining are compared against each other. Our analysis shows that no scheme outperforms the others for every system setup. It suggests, on the other hand, that the low-energy devices need to operate in an adaptive fashion, by choosing the receiver diversity scheme as a function of the imposed requirements.The second contribution consists of introducing a new tractable approach for modeling and optimizing the energy efficiency of cellular networks. Unlike currently available analytical approaches that provide either simple but meaningless or meaningful but complex analytical expressions of the coverage probability and spectral efficiency of cellular networks, the proposed approach is conveniently formulated in a closed-form expression that is proved to be simple and meaningful at the same time. By relying on the new proposed formulation of the spectral efficiency, a new tractable closed-form expression of the energy efficiency of downlink cellular network is proposed, which is used for optimizing the transmit power and the density of cellular base stations. It is mathematically proved, in particular, that the energy efficiency is a unimodal and strictly pseudo-concave function in the transmit power, given the density of the base stations, and in the density of the base stations, given the transmit power. The optimal transmit power and density of base stations are proved to be the solution of simple non-linear equations.The third contribution consists of introducing a new tractable approach for analyzing the performance of multi-tier cellular networks equipped with renewable energy sources, such as solar panels. The proposed approach allows one to account for the spatial distribution of the base stations by using the theory of point processes, as well as for the random arrival and availability of energy by using Markov chain theory. By using the proposed approach, the energy efficiency of cellular networks can be quantified and the interplay between the density of base stations and energy arrival rate can be quantified and optimized.; L'analyse et l'optimisation au niveau de système sont indispensables pour la progression de performance des réseaux de communication. Ils sont nécessaires afin de faire fonctionner de façon optimale des réseaux actuels et de planifier des réseaux futurs. La modélisation et l'analyse au niveau de système des réseaux cellulaires ont été facilitées grâce à la maîtrise de l'outil mathématique de la géométrie stochastique et, plus précisément, la théorie des processus ponctuels spatiaux. Du point de vue de système, il a été empiriquement validé que les emplacements des stations cellulaires de base peuvent être considérés comme des points d'un processus ponctuel de Poisson homogène dont l'intensité coïncide avec le nombre moyen de stations par unité de surface. Dans ce contexte, des contributions de ce travail se trouvent dans le développement de nouvelles méthodologies analytiques pour l'analyse et l'optimisation des déploiements de réseaux cellulaires émergents.La première contribution consiste à introduire une approche pour évaluer la faisabilité de réseaux cellulaires multi-antennes, dans lesquels les dispositifs mobiles à faible énergie décodent les données et récupèrent l'énergie à partir d’un même signal reçu. Des outils de géométrie stochastique sont utilisés pour quantifier le taux d'information par rapport au compromis de puissance captée. Les conclusions montrent que les réseaux d'antennes à grande échelle et les déploiements ultra-denses de stations base sont tous les deux nécessaires pour capter une quantité d'énergie suffisamment élevée et fiable. En outre, la faisabilité de la diversité des récepteurs pour l'application aux réseaux cellulaires descendants est également étudiée. Diverses options basées sur la combinaison de sélection et la combinaison de taux maximal sont donc comparées. Notre analyse montre qu'aucun système n’est plus performant que les autres pour chaque configuration de système : les dispositifs à basse énergie doivent fonctionner de manière adaptative, en choisissant le schéma de diversité des récepteurs en fonction des exigences imposées.La deuxième contribution consiste à introduire une nouvelle approche pour la modélisation et l'optimisation de l'efficacité énergétique des réseaux cellulaires.Contrairement aux approches analytiques actuellement disponibles qui fournissent des expressions analytiques trop simples ou trop complexes de la probabilité de couverture et de l'efficacité spectrale des réseaux cellulaires, l'approche proposée est formulée par une solution de forme fermée qui se révèle en même temps simple et significative. Une nouvelle expression de l'efficacité énergétique du réseau cellulaire descendant est proposée à partir d’une nouvelle formule de l'efficacité spectrale. Cette expression est utilisée pour l’optimisation de la puissance d'émission et la densité des stations cellulaires de base. Il est prouvé mathématiquement que l'efficacité énergétique est une fonction uni-modale et strictement pseudo-concave de la puissance d'émission en fixant la densité des stations de base, et de la densité des stations de base en fixant la puissance d'émission. La puissance d'émission optimale et la densité des stations de base s'avèrent donc être la solution des équations non linéaires simples.La troisième contribution consiste à introduire une nouvelle approche pour analyser les performances des réseaux cellulaires hétérogènes équipés des sources d'énergie renouvelables, telles que les panneaux solaires. L'approche proposée permet de tenir compte de la distribution spatiale des stations de base en utilisant la théorie des processus ponctuels, ainsi que l'apparition aléatoire et la disponibilité de l'énergie en utilisant la théorie des chaînes de Markov. En utilisant l'approche proposée, l'efficacité énergétique des réseaux cellulaires peut être quantifiée et l'interaction entre la densité des stations de base et le taux d'énergie d'apparition peut être quantifiée et optimisée.

In communication networks, system-level analysis and optimization are useful when one is interested in optimizing the system performance across the entire network. System-level analysis and optimization, therefore, are relevant for optimally operating current networks, and for deploying and planning future networks. In the last few years, the system-level modeling and analysis of cellular networks have been facilitated by capitalizing on the mathematical tool of stochastic geometry and, more precisely, on the theory of spatial point processes. It has been empirically validated that, from the system-level standpoint, the locations of cellular base stations can be abstracted as points of a homogeneous Poisson point process whose intensity coincides with the average number of based stations per unit area.In this context, the contribution of the present Ph.D. thesis lies in developing new analytical methodologies for analyzing and optimizing emerging cellular network deployments. The present Ph.D. thesis, in particular, provides three main contributions to the analysis and optimization of energy-efficient cellular networks.The first contribution consists of introducing a tractable approach for assessing the feasibility of multiple-antenna cellular networks, where low-energy mobile devices decode data and harvest power from the same received signal. Tools from stochastic geometry are used to quantify the information rate vs. harvested power tradeoff. Our study unveils that large-scale antenna arrays and ultra-dense deployments of base stations are both necessary to harvest, with high reliability, a sufficiently high amount of power. Furthermore, the feasibility of receiver diversity for application to downlink cellular networks is investigated. Several options that are based on selection combining and maximum ratio combining are compared against each other. Our analysis shows that no scheme outperforms the others for every system setup. It suggests, on the other hand, that the low-energy devices need to operate in an adaptive fashion, by choosing the receiver diversity scheme as a function of the imposed requirements.The second contribution consists of introducing a new tractable approach for modeling and optimizing the energy efficiency of cellular networks. Unlike currently available analytical approaches that provide either simple but meaningless or meaningful but complex analytical expressions of the coverage probability and spectral efficiency of cellular networks, the proposed approach is conveniently formulated in a closed-form expression that is proved to be simple and meaningful at the same time. By relying on the new proposed formulation of the spectral efficiency, a new tractable closed-form expression of the energy efficiency of downlink cellular network is proposed, which is used for optimizing the transmit power and the density of cellular base stations. It is mathematically proved, in particular, that the energy efficiency is a unimodal and strictly pseudo-concave function in the transmit power, given the density of the base stations, and in the density of the base stations, given the transmit power. The optimal transmit power and density of base stations are proved to be the solution of simple non-linear equations.The third contribution consists of introducing a new tractable approach for analyzing the performance of multi-tier cellular networks equipped with renewable energy sources, such as solar panels. The proposed approach allows one to account for the spatial distribution of the base stations by using the theory of point processes, as well as for the random arrival and availability of energy by using Markov chain theory. By using the proposed approach, the energy efficiency of cellular networks can be quantified and the interplay between the density of base stations and energy arrival rate can be quantified and optimized.; L'analyse et l'optimisation au niveau de système sont indispensables pour la progression de performance des réseaux de communication. Ils sont nécessaires afin de faire fonctionner de façon optimale des réseaux actuels et de planifier des réseaux futurs. La modélisation et l'analyse au niveau de système des réseaux cellulaires ont été facilitées grâce à la maîtrise de l'outil mathématique de la géométrie stochastique et, plus précisément, la théorie des processus ponctuels spatiaux. Du point de vue de système, il a été empiriquement validé que les emplacements des stations cellulaires de base peuvent être considérés comme des points d'un processus ponctuel de Poisson homogène dont l'intensité coïncide avec le nombre moyen de stations par unité de surface. Dans ce contexte, des contributions de ce travail se trouvent dans le développement de nouvelles méthodologies analytiques pour l'analyse et l'optimisation des déploiements de réseaux cellulaires émergents.La première contribution consiste à introduire une approche pour évaluer la faisabilité de réseaux cellulaires multi-antennes, dans lesquels les dispositifs mobiles à faible énergie décodent les données et récupèrent l'énergie à partir d’un même signal reçu. Des outils de géométrie stochastique sont utilisés pour quantifier le taux d'information par rapport au compromis de puissance captée. Les conclusions montrent que les réseaux d'antennes à grande échelle et les déploiements ultra-denses de stations base sont tous les deux nécessaires pour capter une quantité d'énergie suffisamment élevée et fiable. En outre, la faisabilité de la diversité des récepteurs pour l'application aux réseaux cellulaires descendants est également étudiée. Diverses options basées sur la combinaison de sélection et la combinaison de taux maximal sont donc comparées. Notre analyse montre qu'aucun système n’est plus performant que les autres pour chaque configuration de système : les dispositifs à basse énergie doivent fonctionner de manière adaptative, en choisissant le schéma de diversité des récepteurs en fonction des exigences imposées.La deuxième contribution consiste à introduire une nouvelle approche pour la modélisation et l'optimisation de l'efficacité énergétique des réseaux cellulaires.Contrairement aux approches analytiques actuellement disponibles qui fournissent des expressions analytiques trop simples ou trop complexes de la probabilité de couverture et de l'efficacité spectrale des réseaux cellulaires, l'approche proposée est formulée par une solution de forme fermée qui se révèle en même temps simple et significative. Une nouvelle expression de l'efficacité énergétique du réseau cellulaire descendant est proposée à partir d’une nouvelle formule de l'efficacité spectrale. Cette expression est utilisée pour l’optimisation de la puissance d'émission et la densité des stations cellulaires de base. Il est prouvé mathématiquement que l'efficacité énergétique est une fonction uni-modale et strictement pseudo-concave de la puissance d'émission en fixant la densité des stations de base, et de la densité des stations de base en fixant la puissance d'émission. La puissance d'émission optimale et la densité des stations de base s'avèrent donc être la solution des équations non linéaires simples.La troisième contribution consiste à introduire une nouvelle approche pour analyser les performances des réseaux cellulaires hétérogènes équipés des sources d'énergie renouvelables, telles que les panneaux solaires. L'approche proposée permet de tenir compte de la distribution spatiale des stations de base en utilisant la théorie des processus ponctuels, ainsi que l'apparition aléatoire et la disponibilité de l'énergie en utilisant la théorie des chaînes de Markov. En utilisant l'approche proposée, l'efficacité énergétique des réseaux cellulaires peut être quantifiée et l'interaction entre la densité des stations de base et le taux d'énergie d'apparition peut être quantifiée et optimisée.

For many measure preserving dynamical systems $(\Omega,T,m)$ the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new process obtained from recording not only the successive times $n$ of visits to a set $A$, but also the position $T^n(x)$ in $A$ of the orbit, in the limit where $m(A)\to0$. We obtain a convergence of this process, suitably normalized, to a Poisson point process in time and space under some decorrelation condition. We present several new applications to hyperbolic maps and SRB measures, including the case of a neighborhood of a periodic point, and some billiards such as Sinai billiards, Bunimovich stadium and diamond billiard.

Achieve faster and more efficient network design and optimization with this comprehensive guide. Some of the most prominent researchers in the field explain the very latest analytic techniques and results from stochastic geometry for modelling the signal-to-interference-plus-noise ratio (SINR) distribution in heterogeneous cellular networks. This book will help readers to understand the effects of combining different system deployment parameters on key performance indicators such as coverage and capacity, enabling the efficient allocation of simulation resources. In addition to covering results for network models based on the Poisson point process, this book presents recent results for when non-Poisson base station configurations appear Poisson, due to random propagation effects such as fading and shadowing, as well as non-Poisson models for base station configurations, with a focus on determinantal point processes and tractable approximation methods. Theoretical results are illustrated with practical Long-Term Evolution (LTE) applications and compared with real-world deployment results.

International audience; The statistical modelling of spatial extremes has recently made major advances. Much of its focus so far has been on the modelling of the magnitudes of extreme events but little attention has been paid on the timing of extremes. To address this gap, this paper introduces the notion of extremal concurrence. Suppose that one measures precipitation at several synoptic stations over multiple days. We say that extremes are concurrent if the maximum precipitation over time at each station is achieved simultaneously, e.g., on a single day. Under general conditions, we show that the finite sample concurrence probability converges to an asymptotic quantity, deemed extremal concurrence probability. Using Palm calculus, we establish general expressions for the extremal concurrence probability through the max-stable process emerging in the limit of the componentwise maxima of the sample. Explicit forms of the extremal concurrence probabilities are obtained for various max-stable models and several estimators are introduced. In particular, we prove that the pairwise extremal concurrence probability for max-stable vectors is precisely equal to the Kendall's $\tau$. The estimators are evaluated by using simulations and applied to study the concurrence patterns of temperature extremes in the United States. The results demonstrate that concurrence probability can provide a powerful new perspective and tools for the analysis of the spatial structure and impact of extremes.

Let [Formula: see text] be a planar Poisson point process of intensity [Formula: see text]. We give a new proof that the expected length of the Voronoi path between [Formula: see text] and [Formula: see text] in the Delaunay triangulation associated with [Formula: see text] is [Formula: see text] when [Formula: see text] goes to infinity; and we also prove that the variance of this length is [Formula: see text]. We investigate the length of possible shortcuts in this path, and define a shortened Voronoi path whose expected length can be expressed as an integral that is numerically evaluated to [Formula: see text]. The shortened Voronoi path has the property to be locally defined; and is shorter than the previously known locally defined paths in Delaunay triangulation such as the upper path whose expected length is [Formula: see text].

International audience; Let (T, V) be a branching random walk on the real line. The extremal process of the branching random walk is the point process of the position of particles at time n shifted by the position of the minimum. Madaule [Mad15] proved that this point process converges toward a shifted decorated Poisson point process. In this article we study the joint convergence of the extremal process with its genealogy informations. This result is then used to characterize the law of the decoration in the limiting process as well as to study the supercritical Gibbs measures of the branching random walk.

LetX:=Xn∪ {(0, 0), (1, 0)}, whereXnis a planar Poisson point process of intensityn. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated withXwhen the intensity ofXngoes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

Let $X_n$ be a $d$ dimensional Poisson point process of intensity $n$. We prove that the expected length of the Voronoi path between two points at distance 1 in the Delaunay triangulation associated with $X_n$ is $\sqrt{\frac{2d}{\pi}}+O(d^{-\frac{1}{2}})$ for all $n\in\mathbb{N}$ and $d\rightarrow\infty$. In any dimension, we provide a precise interval containing the exact value, in 3D the expected length is between 1.4977 and 1.50007.

The main purpose of this thesis is to develop statistical methodologies for stochastic processes data and more precisely Cox process data.The problems considered arise from three different contexts: nonparametric tests, nonparametric kernel estimation and minimax estimation.We first study the statistical test problem of detecting wether a Cox process is Poisson or not.Then, we introduce a semiparametric estimate of the regression over a Poisson point process. Using Itô’s famous chaos expansion for Poisson functionals, we derive asymptotic minimax properties of our estimator.Finally, we introduce a nonparametric estimate of the intensity of a Cox process whenever it is a deterministic function of a known coprocess., L’objectif principal de cette thèse est de développer des méthodologies statistiques adaptées au traitement de données issues de processus stochastiques et plus précisément de processus de Cox.Les problématiques étudiées dans cette thèse sont issues des trois domaines statistiques suivants : les tests non paramétriques, l’estimation non paramétrique à noyaux et l’estimation minimax.Dans un premier temps, nous proposons, dans un cadre fonctionnel, des statistiques de test pour détecter la nature Poissonienne d’un processus de Cox.Nous étudions ensuite le problème de l’estimation minimax de la régression sur un processus de Poisson ponctuel. En se basant sur la décomposition en chaos d’Itô, nous obtenons des vitesses comparables à celles atteintes pour le cas de la régression Lipschitz en dimension finie.Enfin, dans le dernier chapitre de cette thèse, nous présentons un estimateur non-paramétrique de l’intensité d’un processus de Cox lorsque celle-ci est une fonction déterministe d’un co-processus.

International audience; In this paper, a new mathematical approach for the analysis and optimization of cellular-enabled low-energy mobile devices with simultaneous wireless information and power transfer capabilities is introduced. The proposed methodology relies on modeling the locations of the base stations as points of a spatial Poisson point process, and it leverages stochastic geometry for system-level analysis. The tradeoffs emerging from simultaneous wireless information and power transfer transmission are characterized through the concept of “feasibility regions” and are quantified through the joint cumulative distribution function of harvested power and rate. To gain insight on the achievable performance, in addition, an upper bound is proposed, and its accuracy is discussed. The system model encompasses a realistic channel model that accounts for line-of-sight (LOS) and non-LOS (NLOS) links, different cell association criteria, practical receivers based on time switching and power splitting schemes, and directional beamforming. The analysis shows that optimal values for the time switching and power splitting ratios exist and that directional beamforming and network densification are capable of enhancing system performance. More specifically, high directional antennas lead cellular networks to operate in the noise-limited regime, which is proved to provide optimal performance, and because of the existence of LOS and NLOS links, an optimal deployment density for the base stations is proven to exist for typical system setups.

In this paper we consider a continuous almost periodically correlated process $\{X(t),t\in\mathbb{R}\}$ that is observed at the jump moments of a stationary Poisson point process $\{N(t),t\geq0\}$. The processes $\{X(t),t\in\mathbb{R}\}$ and $\{N(t),t\geq0\}$ are assumed to be independent. We define the kernel estimators of the Fourier coefficients of the autocovariance function of $X(t)$ and investigate their asymptotic properties. Moreover, we propose a bootstrap method that provides consistent pointwise and simultaneous confidence intervals for the considered coefficients. Finally, to illustrate our results we provide a simulated data example.

For a Poisson point process X, Itô’s famous chaos expansion implies that every square integrable regression function r with covariate X can be decomposed as a sum of multiple stochastic integrals called chaos. In this paper, we consider the case where r can be decomposed as a sum of δ chaos. In the spirit of Cadre and Truquet [ESAIM: PS 19 (2015) 251–267], we introduce a semiparametric estimate of r based on i.i.d. copies of the data. We investigate the asymptotic minimax properties of our estimator when δ is known. We also propose an adaptive procedure when δ is unknown.

International audience; Assume that several competing methods are available to estimate a parameter in a given statistical model. The aim of estimator averaging is to provide a new estimator, built as a linear combination of the initial estimators, that achieves better properties, under the quadratic loss, than each individual initial estimator. This contribution provides an accessible and clear overview of the method, and investigates its performances on standard spatial point process models. It is demonstrated that the average estimator clearly improves on standard procedures for the considered models. For each example, the code to implement the method with the R software (which only consists of few lines) is provided.

Stein’s method constitutes one of the main techniques to solve some approximation problems in probability theory. In this manuscript, we apply it in the context of point processes. The first part of these investigations focuses on the Poisson point process. Its characteristic independence property provides a way to explain intuitively why a sequence of point processes becoming less and less repulsive can converge to such a point process. More generally, this leads to show some convergence results for some sequences of point processes built by several operations such as superposition, thinning and rescaling. The use of a distance on point processes, the so-called Kantorovich-Rubinstein distance, enables moreover the getting of some convergence rates. The second part is centered on a class of point processes with important attractiveness, called discrete α-stable point processes. Their structure based on a Poisson point process gives us a way to enlarge to these point processes the method used previously and to propose new results, via some properties that we state on these point processes., La méthode de Stein constitue une des principales techniques pour la résolution de certains problèmes d’approximation en théorie des probabilités. Dans ce manuscrit, nous l’appliquons au contexte des processus ponctuels. La première partie de ces investigations se concentre sur le processus ponctuel de Poisson. Sa propriété caractéristique d’indépendance fournit le moyen d’expliquer intuitivement pourquoi une suite de processus ponctuels de moins en moins répulsive peut converger vers un tel processus ponctuel. Ceci nous amène plus généralement à démontrer des résultats de convergence pour des suites de processus ponctuels construites à partir d’opérations telles que la superposition, l’amincissement ou l’homothétie. L’utilisation d’une distance sur les processus ponctuels, appelée distance de Kantorovich-Rubinstein, permet en outre l’obtention de taux de convergence. La seconde partie est centrée sur une classe de processus ponctuels avec beaucoup d’attractivité, appelés processus ponctuels α-stables. Leur structure basée sur un processus ponctuel de Poisson nous permet d’élargir à ces processus la méthode utilisée précédemment et de proposer de nouveaux résultats, via certaines propriétés que nous établissons sur ces processus ponctuels.

Keeler, Ross and Xia (2016) recently derived approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modelled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with different locations and random propagation effects.The aim of this note is to apply some of the main results of Keeler, Ross and Xia (2016) in a less general but more easily applicable form to illustrate how the results can be applied in practice. New results are derived that show that it is the strongest signals, after being weakened by random propagation effects, that behave like a Poisson process, which supports recent experimental work., Comment: 7 pages with 1.5 line spacing

—In this paper, the influence of network deployment on downlink dimensioning under an interference limited regime is studied. Theoretical and numerical results are derived for a network deployed as a Poisson point process. We show that slightly degraded propagation conditions has a beneficial effect on network dimensioning and performance. Spatial influence is also studied thanks to the β-Ginibre model. Simulations show that a regularly deployed network improves dramatically spectrum usage and network performances.

International audience; In this paper, we study a two-way relaying system with analog network coding whereby both the relay and the two sources are under the effect of erroneous channel state information (CSI) and network interference that originate from a Poisson field of interferers. Our contribution is twofold: 1) We characterize the interference affecting the aforementioned nodes by presenting a general model that incorporates the expected correlation of the interference process that affects the relay, which in turn affects the two sources; 2) by deriving the interference statistics, particularly its first moment, we also derive analytical formulas for the distribution of the signal-to-interference-plus-noise ratio (SINR) of the two sources in the high-SINR region, which reveal that in that region, the amount of correlation between the interference processes affecting the relay and the two sources does not affect the SINR distribution. These statistics are subsequently used to derive analytical expressions of the average sum symbol error rate (SER) and the achievable average sum rate of the system. Finally, the theoretical results are corroborated by means of Monte Carlo simulations.

22pp; International audience; This survey will appear as a chapter of the forthcoming book [19]. A U-statistic of order $k$ with kernel $f:\X^k \to \R^d$ over a Poisson process is defined in \cite{ReiSch11} as$$ \sum_{x_1, \dots , x_k \in \eta^k_{\neq}} f(x_1, \dots, x_k) $$ under appropriate integrability assumptions on $f$.U-statistics play an important role in stochastic geometry since many interesting functionals can be written as U-statistics, like intrinsic volumes of intersection processes, characteristics of random geometric graphs, volumes of random simplices, and many others, see for instance \cite{ LacPec13, LPST,ReiSch11}. It turns out that the Wiener-Ito chaos expansion of a U-statistic is finite and thus Malliavin calculus is a particularly suitable method. Variance estimates, the approximation of the covariance structure and limit theorems which have been out of reach for many years can be derived. In this chapter we state the fundamental properties of U-statistics and investigate moment formulae. The main object of the chapter is to introduce the available limit theorems.

We consider a hierarchical network that consists of mobile users, a two-tiered cellular network (namely small cells and macro cells) and central routers, each of which follows a Poisson point process (PPP). In this scenario, small cells with limited-capacity backhaul are able to cache content under a given set of randomized caching policies and storage constraints. Moreover, we consider three different content popularity models, namely fixed content popularity, distance-dependent and load-dependent, in order to model the spatio-temporal behavior of users' content request patterns. We derive expressions for the average delay of users assuming perfect knowledge of content popularity distributions and randomized caching policies. Although the trend of the average delay for all three content popularity models is essentially identical, our results show that the overall performance of cached-enabled heterogeneous networks can be substantially improved, especially under the load-dependent content popularity model., Comment: to be presented at IEEE SPAWC'2016, Edinburgh, UK

International audience; In this paper, we introduce a mathematical approach for system-level analysis and optimization of densely deployed multiple-antenna cellular networks, where low-energy devices are capable of decoding information data and harvesting power simultaneously. The base stations are assumed to be deployed according to a Poisson point process and tools from stochastic geometry are exploited to quantify the trade-off in terms of information rate and harvested power. It is shown that multiple-antenna transmission is capable of increasing information rate and harvested power at the same time.

International audience; We consider a continuum percolation model on $\R^d$, where $d\geq 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Mart\'{i}nez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the critical time when the radius $r$ converges to $0$. The latter does not seem to be deducible from simple scaling arguments. We prove that for $d\geq 4$, there is a positive constant $c$ such that$c^{-1}\sqrt{\log(1/r)}\leq t_c(r)\leq c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t_c(r) \leq c\ r^{(4-d)/2}$ when $d\geq 5$, as $r$ converges to $0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.

We consider a continuum percolation model on $$\mathbb {R}^d$$ , $$d\ge 1$$ . For $$t,\lambda \in (0,\infty )$$ and $$d\in \{1,2,3\}$$ , the occupied set is given by the union of independent Brownian paths running up to time t whose initial points form a Poisson point process with intensity $$\lambda >0$$ . When $$d\ge 4$$ , the Brownian paths are replaced by Wiener sausages with radius $$r>0$$ . We establish that, for $$d=1$$ and all choices of t, no percolation occurs, whereas for $$d\ge 2$$ , there is a non-trivial percolation transition in t, provided $$\lambda $$ and r are chosen properly. The last statement means that $$\lambda $$ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when $$d\in \{2,3\}$$ , but finite and dependent on r when $$d\ge 4$$ ). We further show that for all $$d\ge 2$$ , the unbounded cluster in the supercritical phase is unique. Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest. The present paper settles the basic properties of the model and should be viewed as a springboard for finer results.

The intensity matching approach for tractable performance evaluation and optimization of cellular networks is introduced. It assumes that the base stations are modeled as points of a Poisson point process and leverages stochastic geometry for system-level analysis. Its rationale relies on observing that system-level performance is determined by the intensity measure of transformations of the underlaying spatial Poisson point process. By approximating the original system model with a simplified one, whose performance is determined by a mathematically convenient intensity measure, tractable yet accurate integral expressions for computing area spectral efficiency and potential throughput are provided. The considered system model accounts for many practical aspects that, for tractability, are typically neglected, e.g., line-of-sight and non-line-of-sight propagation, antenna radiation patterns, traffic load, practical cell associations, general fading channels. The proposed approach, more importantly, is conveniently formulated for unveiling the impact of several system parameters, e.g., the density of base stations and blockages. The effectiveness of this novel and general methodology is validated with the aid of empirical data for the locations of base stations and for the footprints of buildings in dense urban environments., Comment: Submitted for Journal Publication

International audience; This paper focuses on the risk of loan default from the point of view of an insurer required to indemnify a bank for losses resulting from a borrower defaulting. The main objective of this paper is to model the provision (or claim reserve) against the risk of borrowers defaulting. Unlike traditionally used models, our model depends on specific information concerning the borrowers (amount borrowed and term of loan). Our approach will also take into account three kinds of dependence: the dependence between each claim amount (by taking into account the real estate price), the dependence between the date of default and the claim amount, and the dependence between the number of defaults and the claim amount. Both theoretical and applied, our model allows the calculation of the mean, the variance, and the law of the provision. The amount of data available allows us to estimate all the parameters and to calculate the mean and the variance plus the quantiles of the provision.

The main focus of this thesis is on performance analysis and system optimization of Long Term Evolution - Advanced (LTE-A) cellular networks by using stochastic geometry. Mathematical analysis of cellular networks is a long-lasting difficult problem. Modeling the network elements as points in a Poisson Point Process (PPP) has been proven to be a tractable yet accurate approach to the performance analysis in cellular networks, by leveraging the powerful mathematical tools such as stochastic geometry. In particular, relying on the PPP-based abstraction model, this thesis develops the mathematical frameworks to the computations of important performance measures such as error probability, coverage probability and average rate in several application scenarios in both uplink and downlink of LTE-A cellular networks, for example, multi-antenna transmissions, heterogeneous deployments, uplink power control schemes, etc. The mathematical frameworks developed in this thesis are general enough and the accuracy has been validated against extensive Monte Carlo simulations. Insights on performance trends and system optimization can be done by directly evaluating the formulas to avoid the time-consuming numerical simulations.; L’objectif principal de cette thèse est l’analyse des performances des réseaux LTE-A (Long Term Evolution- Advanced) au travers de la géométrie stochastique. L’analyse mathématique des réseaux cellulaires est un problème difficile, pour lesquels ils existent déjà un certain nombre de résultats mais qui demande encore des efforts et des contributions sur le long terme. L’utilisation de la géométrie aléatoire et des processus ponctuels de Poisson (PPP) s’est avérée être une approche permettant une modélisation pertinente des réseaux cellulaires et d’une complexité faible (tractable). Dans cette thèse, nous nous intéressons tout particulièrement à des modèles s’appuyant sur ces processus de Poisson : PPP-based abstraction. Nous développons un cadre mathématique qui permet le calcul de quantités reflétant les performances des réseaux LTE-A, tels que la probabilité d’erreur, la probabilité et le taux de couverture, pour plusieurs scénarios couvrant entre autres le sens montant et descendant. Nous considérons également des transmissions multi-antennes, des déploiements hétérogènes, et des systèmes de commande de puissance de la liaison montante. L’ensemble de ces propositions a été validé par un grand nombre de simulations. Le cadre mathématique développé dans cette thèse se veut général, et doit pouvoir s’appliquer à un nombre d’autres scénarios importants. L’intérêt de l’approche proposée est de permettre une évaluation des performances au travers de l’évaluation des formules, et permettent en conséquences d’éviter des simulations qui peuvent prendre énormément de temps en terme de développement ou d’exécution.

International audience; The performance of cellular system significantlydepends on its network topology while cellular networks areundergoing a heterogeneous evolution. This promising trendintroduces unplanned deployment of smaller base stations (BSs),thus complicating the performance evaluation even further. Inthis paper, based on large amount of real BS locations data,we present a comprehensive analysis on the spatial modeling ofcellular network structure. Unlike the related works, we dividethe BSs into different subsets according to geographical factor(e.g. urban or rural) and functional type (e.g. macrocells ormicrocells), and perform detailed spatial analysis to each subset.After recovering the inaccuracy of the Poisson point process(PPP) in BS locations modeling, we take into account the Gibbspoint processes as well as Neyman-Scott point processes andcompare their performance in view of large-scale modeling test.Finally, we turn to the BS quantitative distribution analysis,and reveal the general clustering nature of BSs deployment,which distinctly violates the traditional assumption. Specifically,the -Stable distribution can most precisely reproduce the BSdensity among the popular candidate distributions. This papercarries out a first large-scale identification regarding availableliteratures, and provides more realistic and general results tocontribute to the performance analysis for the forthcomingheterogeneous cellular networks.

Based on a stationary Poisson point process, a wireless network model with random propagation effects (shadowing and/or fading) is considered in order to examine the process formed by the signal-to-interference-plus-noise ratio (SINR) values experienced by a typical user with respect to all the base stations in the down-link channel. This SINR process is completely characterized by deriving its factorial moment measures, which involve numerically tractable, explicit integral expressions. This novel framework naturally leads to expressions for the $k$ -coverage probability, including the case of random SINR threshold values considered in multi-tier network models. While the $k$ -coverage probabilities correspond to the marginal distributions of the order statistics of the SINR process, a more general relation is presented, connecting the factorial moment measures of the SINR process to the joint densities of these order statistics. This gives a way for calculating the exact values of the coverage probabilities arising in a general scenario of signal combination and interference cancellation between base stations. The presented framework consisting of the mathematical representations of SINR characteristics with respect to the factorial moment measures holds for the whole domain of SINR, and is amenable to considerable model extension.