arXiv subjects

Laurent Decreusefond

Publications and source records attributed to Laurent Decreusefond.

At least 19 recordsLinked to original sources

arXiv2026

Statistically Robust Resource Block Allocation for Satellite Communications

It is critical to dimension (accurately estimate capacity of) a satellite system prior to deployment, as it is very expensive to reconfigure launched satellite systems that fail to meet demand or that waste capacity. The fundamental requirement is a dimensioning rule for resource blocks (RBs) given a satellite footprint and a target overload probability (target Quality-of-Service). The rule must be robust to the spatial covariance structure of signal attenuation, which is generally unknown both at the time of pre-deployment dimensioning and afterwards. Existing approaches address parts of this problem, but there does not yet exist a footprint-level RB dimensioning rule for the satellite context. We develop such a rule: starting with a Gaussian attenuation field that induces a covariance structure inspired by classical work on spatial covariance of attenuation, we sample users at random along with their field-based attenuation values, and estimate aggregate RB demand for a target overload probability. We do this in two complementary ways: a Monte Carlo route that gives a simulation-derived RB budget for a given target overload probability, and a concentration route that gives a conservative analytic upper bound on the target overload probability for a given RB budget (such as the one obtained through simulation). Taken together, these complementary approaches give a principled way to dimension RBs for a satellite footprint under spatially correlated attenuation.

cs.NI↗

arXiv2026

Rate of convergence of the conditioned random walk towards the Brownian bridge

We study the rate of convergence of two discrete processes towards the Brownian bridge: the random walk conditioned to be zero at time 2n and the empirical process which appears in the Glivencko-Cantelli theorem. Combining a functional Stein method with a Radon-Nikodym representation of the bridge, we bound the Fortet-Mourier distance between these conditioned processes and the Brownian bridge.

math.PR↗

arXiv2025

Quantitative Limit Theorems for Cox-Poisson and Cox-Binomial Point Processes

This paper establishes quantitative limit theorems for two classes of Cox point processes, quantifying their convergence to a Poisson point process (PPP). We employ Stein's method for PPP aproximation, leveraging the generator approach and the Stein-Dirichlet representation formula associated with the Glauber dynamics. First, we investigate a Cox-Poisson process constructed by placing one-dimensional PPPs on the lines of a Poisson line process in $\mathbb{R}^2. We derive an explicit bound on the convergence rate to a homogeneous PPP as the line intensity grows and the point intensity on each line diminishes. Second, we analyze a Cox-Binomial process on the unit sphere $\mathbb{S}^2$, modeling a system of satellites. This process is generated by placing PPPs on great-circle orbits, whose positions are determined by a Binomial point process. For this model, we establish a convergence rate of order $O(1/n)$ to a uniform PPP on the sphere, where n is the number of orbits. The derived bounds provide precise control over the approximation error in both models, with applications in stochastic geometry and spatial statistics.

math.PR↗

arXiv2025

Functional analysis of multivariate max-stable distributions

We study the connections existing between max-infinitely divisible distributions and Poisson processes from the point of view of functional analysis. More precisely, we derive functional identities for the former by using well-known results of Poisson stochastic analysis. We also introduce a family of Markov semigroups whose stationary measures are the so-called multivariate max-stable distributions. Their generators thus provide a functional characterization of extreme valued distributions in any dimension. Additionally, we give a few functional identities associated to those semi-groups, namely a Poincar{\'e} identity and commutation relations. Finally, we present a stochastic process whose semigroup corresponds to the one we introduced and that can be expressed using extremal stochastic integrals.

math.FA↗

arXiv2025

Optimal transport, determinantal point processes and the Bergman kernel

We study the Bergman determinantal point process from a theoretical point of view motivated by its simulation. We construct restricted and restricted-truncated variants of the Bergman kernel and show optimal transport inequalities involving the Kantorovitch-Rubinstein Wasserstein distance to show to what extent it is fair to truncate the restriction of this point process to a compact ball of radius $1 - \varepsilon $. We also investigate the deviation of the number of points of the restricted Bergman determinantal point process, indicate which number of points looks like an optimal choice, and provide upper bounds on its deviation, providing an answer to an open question asked in [5]. We also consider restrictions to other regions and investigate the choice of such regions for restriction. Finally, we provide general results as to the deviation of the number of points of any determinantal point process.

math.PR↗

arXiv2025

Stein's method for max-stable random vectors

Motivated by the omnipresence of extreme value distributions in limit theorems involving extremes of random processes, we adapt Stein's method to include these laws as possible target distributions. We do so by using the generator approach of Stein's method, which is possible thanks to a recently introduced family of semi-groups. We study the corresponding Stein solution and its properties when the working distance is either the smooth Wasserstein distance or the Kolmogorov distance. We make use of those results to bound the distance between two max-stable random vectors, as well as to get a rate of convergence for the de Haan-LePage series in smooth Wasserstein distance.

math.PR↗

arXiv2024

Performance analysis of a RIS-assisted communications

Reconfigurable Intelligent Surfaces (RIS) are currently considered for adoption in future 6G stantards. ETSI and 3GPP have started feasibility and performance investigations of such a technology. This work proposes an analytical model to analyze RIS performance. It relies on a simple street model where obstacles and mobile units are all aligned. RIS is positioned onto a building parallel to the road. The coverage probability in presence of obstacles and concurrent communications is then computed as a performance criteria.

cs.NI↗

arXiv2024

Random walks on simplicial complexes

The notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs, and contains information on the topology of these structures. Even for a graph, the consideration of associated simplicial complexes is interesting to understand its shape. Whereas the Laplacian of a graph has a simple probabilistic interpretation as the generator of a continuous time Markov chain on the graph, things are not so direct when considering simplicial complexes. We define here new Markov chains on simplicial complexes. For a given order~$k$, the state space is the set of $k$-cycles that are chains of $k$-simplexes with null boundary. This new framework is a natural generalization of the canonical Markov chains on graphs. We show that the generator of our Markov chain is the upper Laplacian defined in the context of algebraic topology for discrete structure. We establish several key properties of this new process: in particular, when the number of vertices is finite, the Markov chain is positive recurrent. This result is not trivial, since the cycles can loop over themselves an unbounded number of times. We study the diffusive limits when the simplicial complexes under scrutiny are a sequence of ever refining triangulations of the flat torus. Using the analogy between singular and Hodge homologies, we express this limit as valued in the set of currents. The proof of tightness and the identification of the limiting martingale problem make use of the flat norm and carefully controls of the error terms in the convergence of the generator. Uniqueness of the solution to the martingale problem is left open. An application to hole detection is carried.

math.PR↗

arXiv2024

Malliavin structure for conditionally independent random variables

On any denumerable product of probability spaces, we extend the discrete Malliavin structure for conditionally independent random variables. As a consequence, we obtain the chaos decomposition for functionals of conditionally independent random variables. We also show how to derive some concentration results in that framework. The Malliavin-Stein method yields Berry-Esseen bounds for U-Statistics of such random variables. It leads to quantitative statements of conditional limit theorems: Lyapunov's central limit theorem, De Jong's limit theorem for multilinear forms. The latter is related to the fourth moment phenomenon. The final application consists of obtaining the rates of normal approximation for subhypergraph counts in random exchangeable hypergraphs including the Erd{ö}s-R{é}nyi hypergraph model. The estimator of subhypergraph counts is an example of homogeneous sums for which we derive a new decomposition that extends the Hoeffding decomposition.

math.PR↗

arXiv2024

Rate of Convergence in the Functional Central Limit Theorem for Stable Processes

In this article, we quantify the functional convergence of the rescaled random walk with heavy tails to a stable process.This generalizes the Generalized Central Limit Theorem for stable random variables infinite dimension. We show that provided we have a control between the randomwalk or the limiting stable process and their respective affine interpolation, we canlift the rate of convergence obtained for multivariate distributions to a rateof convergence in some functional spaces.

math.PR↗

arXiv2023

Invertibility of functionals of the Poisson process and applications

Following previous investigations by {\"U}st{\"u}nel [22] about the invertibility of some transformations on the Wiener space, we find some entropic conditions under which a random change of time is invertible on the Poisson space. As a consequence, we provide a new construction of Hawkes processes. We also establish a new variational representation of the entropy.

math.PR↗

arXiv2021

Diffusive limits of Lipschitz functionals of Poisson measures

Continuous Time Markov Chains, Hawkes processes and many other interesting processes can be described as solution of stochastic differential equations driven by Poisson measures. Previous works, using the Stein's method, give the convergence rate of a sequence of renormalized Poisson measures towards the Brownian motion in several distances, constructed on the model of the Kantorovitch-Rubinstein (or Wasserstein-1) distance. We show that many operations (like time change, convolution) on continuous functions are Lipschitz continuous to extend these quantified convergences to diffuse limits of Markov processes and long-time behavior of Hawkes processes.

math.PR↗

arXiv2020

Optimal transport between determinantal point processes and application to fast simulation

We analyze several optimal transportation problems between de-terminantal point processes. We show how to estimate some of the distances between distributions of DPP they induce. We then apply these results to evaluate the accuracy of a new and fast DPP simulation algorithm. We can now simulate in a reasonable amount of time more than ten thousands points.

cs.DS↗

arXiv2020

Stein's method for diffusive limit of queueing processes

Donsker Theorem is perhaps the most famous invariance principle result for Markov processes. It states that when properly normalized, a random walk behaves asymptotically like a Brownian motion. This approach can be extended to general Markov processes whose driving parameters are taken to a limit, which can lead to insightful results in contexts like large distributed systems or queueing networks. The purpose of this paper is to assess the rate of convergence in these so-called diffusion approximations, in a queueing context. To this end, we extend the functional Stein method introduced for the Brownian approximation of Poisson processes, to two simple examples: the single-server queue and the infinite-server queue. By doing so, we complete the recent applications of Stein's method to queueing systems, with results concerning the whole trajectory of the considered process, rather than its stationary distribution.

math.PR↗

arXiv2020

Construction and Random Generation of Hypergraphs with Prescribed Degree and Dimension Sequences

We propose algorithms for construction and random generation of hypergraphs without loops and with prescribed degree and dimension sequences. The objective is to provide a starting point for as well as an alternative to Markov chain Monte Carlo approaches. Our algorithms leverage the transposition of properties and algorithms devised for matrices constituted of zeros and ones with prescribed row- and column-sums to hypergraphs. The construction algorithm extends the applicability of Markov chain Monte Carlo approaches when the initial hypergraph is not provided. The random generation algorithm allows the development of a self-normalised importance sampling estimator for hypergraph properties such as the average clustering coefficient.We prove the correctness of the proposed algorithms. We also prove that the random generation algorithm generates any hypergraph following the prescribed degree and dimension sequences with a non-zero probability. We empirically and comparatively evaluate the effectiveness and efficiency of the random generation algorithm. Experiments show that the random generation algorithm provides stable and accurate estimates of average clustering coefficient, and also demonstrates a better effective sample size in comparison with the Markov chain Monte Carlo approaches.

cs.DS↗

arXiv2020

Dynamic-TDD Interference Tractability Approaches and Performance Analysis in Macro-Cell and Small-Cell Deployments

Meeting the continued growth in data traffic volume, Dynamic Time Division Duplex (D-TDD) has been introduced as a solution to deal with the uplink (UL) and downlink (DL) traffic asymmetry, mainly observed for dense heterogeneous network deployments, since it is based on instantaneous traffic estimation and provide more flexibility in resource assignment. However, the use of this feature requires new interference mitigation schemes capable to handle two additional types of interference between cells in opposite transmission direction: DL to UL and UL to DL interference. The aim of this work is to provide a complete analytical approach to model inter-cell interference in macro-cell and dense small-cell networks. We derive the explicit expressions of Interference to Signal Ratio (ISR) at each position of the network, in both DL and UL, to quantify the impact of each type of interference on the system performance. Also, we provide the explicit expressions of the coverage probability as functions of different system parameters by covering different scenarios. Finally, through system level simulations, we analyze the feasibility of D-TDD implementation in both deployments and we compare its performance to the static-TDD (S-TDD) configuration.

cs.NI↗

arXiv2020

A 3D Beamforming Scheme Based on The Spatial Distribution of User Locations

Multi-antenna technologies such as massive Multiple-Input Multiple-Output (massive MIMO) and beamforming are key features to enhance performance, in terms of capacity and coverage, by using a large number of antennas intelligently. With the upcoming 5G New Radio (NR), FD-MIMO (Full Dimension MIMO) will play a major key role. FD-MIMO consists in arranging a large number of antennas in a 2D array, which enables to use 3D beamforming i.e., beamforming in both horizontal and vertical dimensions. The present paper provides a 3D beamforming model where beam steering depends on the random spatial distribution of users. We attempt to derive some analytical results regarding the probability distribution of antenna beamforming radiation pattern. Also, through system level simulations, we show how 3D beamforming can reduce interference impact, compared to the traditional 2D beamforming, and enhances system performance in terms of the coverage probability and users throughput.

cs.NI↗

arXiv2020

How To Dimension Radio Resources When Users Are Distributed on Roads Modeled by Poisson Line Process

Resources dimensioning aims at finding the number of radio resources required to carry a forecast data traffic at a target users Quality of Services (QoS). The present paper attempts to provide a new approach of radio resources dimensioning considering the congestion probability, qualified as a relevant metric for QoS evaluation. Users are assumed to be distributed according to a linear Poisson Point Process (PPP) in a random system of roads modeled by Poisson Line Process (PLP) instead of the widely-used spatial PPP. We derive the analytical expression of the congestion probability for analyzing its behavior as a function of network parameters. Finally we show how to dimension radio resources by setting a value of the congestion probability, often targeted by the operator, in order to find the relation between the necessary resources and the forecast data traffic expressed in terms of cell throughput. Different numerical results are presented to justify this dimensioning approach.

cs.NI↗