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27 results on '"Laboratoire de Mathématiques"'

1. Change-level detection for Lévy subordinators

Author

Zeina Al Masry, Landy Rabehasaina, Ghislain Verdier, FEMTO-ST institute, Université de Bourgogne Franche-Comté, CNRS, ENSMM, Besançon, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), and Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Statistics and Probability, Lévy process, sequential testing, Applied Mathematics, Modeling and Simulation, CUSUM, Change detection, subordinator
Abstract

Let $\boldsymbol{X}=(X_t)_{t\ge 0}$ be a process behaving as a general increasing Lévy process (subordinator) prior to hitting a given unknown level $m_0$, then behaving as another different subordinator once this threshold is crossed. This paper addresses the detection of this unknown threshold $m_0\in [0,+\infty]$ from an observed trajectory of the process. These kind of model and issue are encountered in many areas such as reliability and quality control in degradation problems. More precisely, we construct, from a sample path and for each $\epsilon >0$, a so-called detection level $L_\epsilon$ by considering a CUSUM inspired procedure. Under mild assumptions, this level is such that, while $m_0$ is infinite (i.e. when no changes occur), its expectation $ \mathbb{E}_{\infty}(L_{\epsilon})$ tends to $+\infty$ as $\epsilon$ tends to $0$, and the expected overshoot $ \mathbb{E}_{m_0}([L_{\epsilon} - m_0]^+)$, while the threshold $m_0$ is finite, is negligible compared to $ \mathbb{E}_{\infty}(L_{\epsilon})$ as $\epsilon$ tends to $0$. Numerical illustrations are provided when the Lévy processes are gamma processes with different shape parameters.

Published
2022
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2. SDEs with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions

Author

Monique Jeanblanc, Frédéric Vrins, Damiano Brigo, UCL - SSH/LIDAM/LFIN - Louvain Finance, UCL - SSH/LIDAM/CORE - Center for operations research and econometrics, Department of Mathematics [Imperial College London], Imperial College London, Laboratoire de Mathématiques et Modélisation d'Evry (LaMME), Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Centre National de la Recherche Scientifique (CNRS), Université Catholique de Louvain = Catholic University of Louvain (UCL), Institut National de la Recherche Agronomique (INRA) - Université d'Evry-Val d'Essonne - ENSIIE - Centre National de la Recherche Scientifique (CNRS), Université Catholique de Louvain (UCL), ANR 11-LABX-0019., Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et Modélisation d'Evry, JEANBLANC, Monique, UCL - SSH/IMMAQ/CORE - Center for operations research and econometrics, and UCL - SSH/IMMAQ/LFIN-Louvain Finance

Subjects
Uniform Martingale Diffusions, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Pure mathematics, Uniformly distributed stochastic differential equation, [MATH] Mathematics [math], math.PR, 01 natural sciences, gales, 010104 statistics & probability, Stochastic differential equation, Mathematics::Probability, 0102 Applied Mathematics, 60J60 * Department of Mathematics, Mean Reverting Uniform Diffusion, Mean reverting uniform diffusion, Maximum entropy stochastic recovery rates, [MATH]Mathematics [math], Maximum Entropy, Stochastic Recovery Rates, Imperial College, Mathematics, Conic Martin-gales, peacock process, Applied Mathematics, Weak solution, Principle of maximum entropy, 0104 Statistics, 1502 Banking, Finance and Investment, Uniformly distributed Stochastic Differential Equation, Uniform SDE Sim-ulation, Conic section, Modeling and Simulation, ulation, Peacock Process, Martingale (probability theory), Statistics and Probability, uniformmly distributed diffusion, Uniformly distributed diffusion, Statistics & Probability, conic martingales, Uniform SDE Sim, Mathematical Finance and Stochastic Analysis groups, Uniform Martingale diffusions, Maximum Entropy Stochastic Correlation, Modelling and Simulation, Uniformly distributed Diffusion, Ergodic theory, Maximum entropy stochastic correlation, 0101 mathematics, 010102 general mathematics, Maximum Entropy Stochastic Recovery Rates, Regular polygon, AMS classification codes: 60H10, mean reverting uniform SDE, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60H10, 60J60, Mean Reverting Uniform SDE, Local time, Conic Martin, Uniform SDE simulation
Abstract

It is known since Kellerer (1972) that for any peacock process there exist mar-tingales with the same marginal laws. Nevertheless, there is no general method for finding such martingales that yields diffusions. Indeed, Kellerer's proof is not constructive: finding the dynamics of processes associated to a given peacock is not trivial in general. In this paper we are interested in the uniform peacock that is, the peacock with uniform law at all times on a generic time-varying support [a(t), b(t)]. We derive explicitly the corresponding Stochastic Differential Equations (SDEs) and prove that, under certain conditions on the boundaries a(t) and b(t), they admit a unique strong solution yielding the relevant diffusion process. We discuss the relationship between our result and the previous derivation of diffusion processes associated to square-root and linear time-boundaries, emphasizing the cases where our approach adds strong uniqueness, and study the local time and activity of the solution processes. We then study the peacock with uniform law at all times on a constant support [−1, 1] and derive the SDE of an associated mean-reverting diffusion process with uniform margins that is not a martingale. For the related SDE we prove existence of a solution in [0, T ]. Finally, we provide a numerical case study showing that these processes have the desired uniform behaviour. These results may be used to model random probabilities, random recovery rates or random correlations.

Published
2020
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3. A boundary driven generalized contact process with exchange of particles: Hydrodynamics in infinite volume

Author

Ellen Saada, Kevin Kuoch, Mustapha Mourragui, Dynamical Systems, Geometry & Mathematical Physics, Johann Bernoulli Institute for Mathematics and Computer Science (JBI), University of Groningen [Groningen], Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0020,LSD,Large Stochastic Dynamical Models in Non-Equilibrium Statistical Physics(2015), Johann Bernoulli Institute for Mathematics and Computer Science, Laboratoire de Mathématiques Raphaël Salem ( LMRS ), Université de Rouen Normandie ( UNIROUEN ), Normandie Université ( NU ) -Normandie Université ( NU ) -Centre National de la Recherche Scientifique ( CNRS ), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), ANR : LSD,ANR-15-CE40-0020-02 - LSD - Large Stochastic Dynamics, Centre National de la Recherche Scientifique (CNRS) - Université de Rouen, MAP5 - Mathématiques Appliquées à Paris 5 (MAP5), Université Paris Descartes - Paris 5 (UPD5) - Institut National des Sciences Mathématiques et de leurs Interactions - Centre National de la Recherche Scientifique (CNRS), Bernoulli Institute for Mathematics and Computer Science and Artificial Intelligence, and ANR-15-CE40-0020,LSD,Modèles stochastiques en grande dimension pour la physique statistique hors équilibre(2015)

Subjects
Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Current, Boundary (topology), Specific entropy, LIMIT, 01 natural sciences, Domain (mathematical analysis), generalised contact process, 010104 statistics & probability, symbols.namesake, Superposition principle, Infinite volume, SYSTEMS, Law of large numbers, Reaction–diffusion system, FOS: Mathematics, Calculus, 0101 mathematics, Two species process, Mathematics, LARGE DEVIATIONS, Contact process, ATTRACTIVENESS, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Mathematical analysis, REACTION DIFFUSION-EQUATIONS, FLUCTUATIONS, MSC: Primary 82C22, Secondary 60F10, 82C35, Stationary nonequilibrium states, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Generalized contact process, Reservoirs, Modeling and Simulation, Dirichlet boundary condition, symbols, Hydrodynamic limit, EXCLUSION PROCESSES, LARGE NUMBERS, Particle, LANDAU LATTICE MODEL, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], LAW, Mathematics - Probability
Abstract

We consider a two species process which evolves in a finite or infinite domain in contact with particles reservoirs at different densities, according to the superposition of a generalised contact process and a rapid-stirring dynamics in the bulk of the domain, and a creation/annihilation mechanism at its boundaries. For this process, we study the law of large numbers for densities and current. The limiting equations are given by a system of non-linear reaction-diffusion equations with Dirichlet boundary conditions., 36 pages; soumis

Published
2017
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4. Martingales and rates of presence in homogeneous fragmentations

Author

Alain Rouault, Nathalie Krell, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques de Versailles ( LMV ), Université Paris-Saclay-Centre National de la Recherche Scientifique ( CNRS ) -Université de Versailles Saint-Quentin-en-Yvelines ( UVSQ ), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Statistics and Probability, Logarithm, 01 natural sciences, Lévy process, Combinatorics, 010104 statistics & probability, Exponential growth, martingales, fragmentation, Modelling and Simulation, FOS: Mathematics, 0101 mathematics, Mathematics, Discrete mathematics, probability tilting, Concave function, Applied Mathematics, 010102 general mathematics, Probability (math.PR), Random walk, Exponential function, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Homogeneous, Modeling and Simulation, A.M.S. 60J85, 65J25, 60G09, Martingale (probability theory), [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability
Abstract

The main focus of this work is the asymptotic behavior of mass-conservative homogeneous fragmentations. Considering the logarithm of masses makes the situation reminiscent of branching random walks. The standard approach is to study {\bf asymptotical} exponential rates. For fixed $v > 0$, either the number of fragments whose sizes at time $t$ are of order $\e^{-vt}$ is exponentially growing with rate $C(v) > 0$, i.e. the rate is effective, or the probability of presence of such fragments is exponentially decreasing with rate $C(v) < 0$, for some concave function $C$. In a recent paper, N. Krell considered fragments whose sizes decrease at {\bf exact} exponential rates, i.e. whose sizes are confined to be of order $\e^{-vs}$ for every $s \leq t$. In that setting, she characterized the effective rates. In the present paper we continue this analysis and focus on probabilities of presence, using the spine method and a suitable martingale., In this version, we correct a misprint in Assumption A from the previous one

Published
2011
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5. On the long time behavior of the TCP window size process

Author

Djalil Chafaï, Katy Paroux, Florent Malrieu, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Dependability Interoperability and perfOrmance aNalYsiS Of networkS (DIONYSOS), Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-RÉSEAUX, TÉLÉCOMMUNICATION ET SERVICES (IRISA-D2), Institut de Recherche en Informatique et Systèmes Aléatoires (IRISA), Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Université de Bretagne Sud (UBS)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Rennes (ENS Rennes)-Institut National de Recherche en Informatique et en Automatique (Inria)-Télécom Bretagne-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-CentraleSupélec-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Université de Rennes (UNIV-RENNES)-CentraleSupélec-Institut de Recherche en Informatique et Systèmes Aléatoires (IRISA), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Université de Bretagne Sud (UBS)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Rennes (ENS Rennes)-Télécom Bretagne-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-CentraleSupélec, ANR-08-BLAN-0242,EVOL,Dissipative Evolutions and Convergence to Equilibrium(2008), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Bretagne Sud (UBS)-École normale supérieure - Rennes (ENS Rennes)-Institut National de Recherche en Informatique et en Automatique (Inria)-Télécom Bretagne-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Bretagne Sud (UBS)-École normale supérieure - Rennes (ENS Rennes)-Institut National de Recherche en Informatique et en Automatique (Inria)-Télécom Bretagne-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche en Informatique et Systèmes Aléatoires (IRISA), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Bretagne Sud (UBS)-École normale supérieure - Rennes (ENS Rennes)-Télécom Bretagne-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Analyse et de Mathématiques Appliquées ( LAMA ), Université Paris-Est Marne-la-Vallée ( UPEM ) -Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 ( UPEC UP12 ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques de Besançon ( LMB ), Centre National de la Recherche Scientifique ( CNRS ) -Université de Franche-Comté ( UFC ), Dependability Interoperability and perfOrmance aNalYsiS Of networkS ( DIONYSOS ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -RÉSEAUX, TÉLÉCOMMUNICATION ET SERVICES ( IRISA-D2 ), Institut de Recherche en Informatique et Systèmes Aléatoires ( IRISA ), CentraleSupélec-Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -Télécom Bretagne-Institut National des Sciences Appliquées ( INSA ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -École normale supérieure - Rennes ( ENS Rennes ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Bretagne Sud ( UBS ) -CentraleSupélec-Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -Télécom Bretagne-Institut National des Sciences Appliquées ( INSA ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -École normale supérieure - Rennes ( ENS Rennes ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Bretagne Sud ( UBS ) -Institut de Recherche en Informatique et Systèmes Aléatoires ( IRISA ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -Télécom Bretagne-Institut National des Sciences Appliquées ( INSA ) -École normale supérieure - Rennes ( ENS Rennes ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Bretagne Sud ( UBS ), ANR-08-BLAN-0242,EVOL,Dissipative Evolutions and Convergence to Equilibrium ( 2008 ), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Bretagne Sud (UBS)-École normale supérieure - Rennes (ENS Rennes)-Institut National de Recherche en Informatique et en Automatique (Inria)-Télécom Bretagne-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université de Rennes 1 (UR1), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Bretagne Sud (UBS)-École normale supérieure - Rennes (ENS Rennes)-Institut National de Recherche en Informatique et en Automatique (Inria)-Télécom Bretagne-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche en Informatique et Systèmes Aléatoires (IRISA), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Bretagne Sud (UBS)-École normale supérieure - Rennes (ENS Rennes)-Télécom Bretagne-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-INSTITUT AGRO Agrocampus Ouest, Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)-Université de Bourgogne (UB), CentraleSupélec-Télécom Bretagne-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure - Rennes (ENS Rennes)-Université de Bretagne Sud (UBS)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-CentraleSupélec-Télécom Bretagne-Université de Rennes 1 (UR1), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Institut de Recherche en Informatique et Systèmes Aléatoires (IRISA), and Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Université de Bretagne Sud (UBS)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes)

Subjects
FOS: Computer and information sciences, Statistics and Probability, Mathematical optimization, Transmission Control Protocol, [ INFO.INFO-NI ] Computer Science [cs]/Networking and Internet Architecture [cs.NI], Additive Increase–Multiplicative Decrease processes (AIMD), Exponential Ergodicity, Markov process, 01 natural sciences, Computer Science - Networking and Internet Architecture, Piecewise linear function, Coupling, [INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI], 010104 statistics & probability, symbols.namesake, Modelling and Simulation, FOS: Mathematics, Additive increase/multiplicative decrease, 68M12, 60K30, 60K25, 90B18, Statistical physics, 0101 mathematics, TCP window scale option, Randomness, Mathematics, Networking and Internet Architecture (cs.NI), Queuing Theory, Stochastic process, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Ergodicity, Network Protocols, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Piecewise Deterministic Markov Processes (PDMP), Queueing theory, Modeling and Simulation, symbols, Additive Increase Multiplicative Decrease Processes (AIMD), [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], 68M12, 60K30, 60K25, 90B18, Mathematics - Probability
Abstract

The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in $[0,\infty)$, is ergodic and irreversible. It belongs to the Additive Increase Multiplicative Decrease class of processes. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. Several aspects of this process have already been investigated in the literature. In the present paper, we mainly get quantitative estimates for the convergence to equilibrium, in terms of the $W_1$ Wasserstein coupling distance, for the process and also for its embedded chain., Comment: Corrections

Published
2010
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6. A constructive approach to Euler hydrodynamics for attractive processes. Application to k-step exclusion

Author

Krishnamurthi Ravishankar, Ellen Saada, Christophe Bahadoran, Hervé Guiol, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Ecole Polytechnique Fédérale de Lausanne (EPFL), Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), and Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, MCS : 60K35, 82C22, non concave non convex flux, attractive particle systems, 01 natural sciences, Constructive, Convexity, k-step exclusion process, 010104 statistics & probability, symbols.namesake, Law of large numbers, Modelling and Simulation, Calculus, Applied mathematics, 0101 mathematics, 10. No inequality, Mathematics, k-step exclusion, Applied Mathematics, 010102 general mathematics, Non-convex or non-concave flux, Invariant (physics), Contact discontinuities, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Riemann hypothesis, Glimm scheme, Modeling and Simulation, Euler's formula, symbols, Hydrodynamics, Invariant measure, Entropy solution, Jump process
Abstract

International audience; We derive by a constructive method the hydrodynamic behavior of attractive processes with irreducible jumps and product invariant measures. Our approach relies on (i) explicit construction of Riemann solutions without assuming convexity, which may lead to contact discontinuities and (ii) a general result which proves that the hydrodynamic limit for Riemann initial profiles implies the same for general initial pro5les. The k-step exclusion process provides a simple example. We also give a law of large numbers for the tagged particle in a nearest neighbor asymmetric k-step exclusion process.

Published
2002
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7. Berry–Esseen bounds and moderate deviations for random walks on GLd(R)

Author

Ion Grama, Quansheng Liu, Hui Xiao, Universitat Hildesheim, Institut fur Mathematik and Angewandte Informatik, Hildesheim, Germany, Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), and Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Independent and identically distributed random variables, September 1, Spectral radius, 2021. 2010 Mathematics Subject Classification. Primary 60F10, Applied Mathematics, 010102 general mathematics, General linear group, 16. Peace & justice, Random walk, 01 natural sciences, Exponential function, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010101 applied mathematics, Combinatorics, 60J05, Modeling and Simulation, Projective space, Irreducibility, 0101 mathematics, Secondary 60B20, Operator norm, Mathematics
Abstract

Let ( g n ) n ⩾ 1 be a sequence of independent and identically distributed random elements of the general linear group G L d ( R ) , with law μ . Consider the random walk G n : = g n … g 1 . Denote respectively by ‖ G n ‖ and ρ ( G n ) the operator norm and the spectral radius of G n . For log ‖ G n ‖ and log ρ ( G n ) , we prove moderate deviation principles under exponential moment and strong irreducibility conditions on μ ; we also establish moderate deviation expansions in the normal range [ 0 , o ( n 1 / 6 ) ] and Berry–Esseen bounds under the additional proximality condition on μ . Similar results are found for the couples ( X n x , log ‖ G n ‖ ) and ( X n x , log ρ ( G n ) ) with target functions, where X n x : = G n ⋅ x is a Markov chain and x is a starting point on the projective space P ( R d ) .

Published
2021
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8. Self-normalized Cramér type moderate deviations for stationary sequences and applications

Author

Qi-Man Shao, Quansheng Liu, Xiequan Fan, Ion Grama, Center for Applied mathematics, Tianjin University, Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), and Southern University of Science and Technology [Shenzhen] (SUSTech)

Subjects
Statistics and Probability, Mathematics::Combinatorics, Applied Mathematics, 010102 general mathematics, Self normalized, Computer Science::Computational Geometry, Type (model theory), Stationary sequence, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Combinatorics, 010104 statistics & probability, Integer, Computer Science::Discrete Mathematics, Modeling and Simulation, 60F10, 60G10, 60E15, Moderate deviations, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics - Probability, Mathematics
Abstract

Let $(X _i)_{i\geq1}$ be a stationary sequence. Denote $m=\lfloor n^\alpha \rfloor, 0< \alpha < 1,$ and $ k=\lfloor n/m \rfloor,$ where $\lfloor a \rfloor$ stands for the integer part of $a.$ Set $S_{j}^\circ = \sum_{i=1}^m X_{m(j-1)+i}, 1\leq j \leq k,$ and $ (V_k^\circ)^2 = \sum_{j=1}^k (S_{j}^\circ)^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbb{P}( \sum_{j=1}^k S_{j}^\circ /V_k^\circ \geq x)$ as $n\to \infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed., Comment: 30 pages

Published
2020
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9. Persistence probabilities for stationary increment processes

Author

Frank Aurzada, Nadine Guillotin-Plantard, Françoise Pène, Technische Universität Darmstadt (TU Darmstadt), Probabilités, statistique, physique mathématique (PSPM), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), Université de Brest (UBO)-Université de Bretagne Sud (UBS)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), and Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Fractional Brownian motion, Stochastic process, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Dynamical Systems (math.DS), Random walk, 01 natural sciences, Upper and lower bounds, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Scaling limit, Mathematics::Probability, Modeling and Simulation, FOS: Mathematics, Range (statistics), Statistical physics, Mathematics - Dynamical Systems, 0101 mathematics, Random variable, Mathematics - Probability, Brownian motion, Mathematics
Abstract

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion, random walks in random sceneries, random processes in Brownian scenery, and the Matheron-de Marsily model in Z^2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of optimal order in the general case and improved lower bounds (compared to previous literature) for many processes., This version corrects some mistakes from the printed version. In particular, the assumptions of Theorem 4 needed to be changed; Theorem 20 needed an extra assumption; the proof of Theorem 11 has been changed

Published
2018
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10. The jamming constant of uniform random graphs

Author

Pascal Moyal, Matthieu Jonckheere, Paola Bermolen, Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), and Université de Technologie de Compiègne (UTC)

Subjects
60J25, 05C80, 60B12, Statistics and Probability, Matemáticas, 01 natural sciences, Matemática Pura, purl.org/becyt/ford/1 [https], Combinatorics, 010104 statistics & probability, 0103 physical sciences, Random regular graph, FOS: Mathematics, [MATH]Mathematics [math], 0101 mathematics, 010306 general physics, ComputingMilieux_MISCELLANEOUS, Mathematics, Random graph, Discrete mathematics, RANDOM GRAPH, Applied Mathematics, Probability (math.PR), purl.org/becyt/ford/1.1 [https], Random function, Random element, MEASURE-VALUED MARKOV PROCESS, Random walk, Random variate, HYDRODYNAMIC LIMIT, Modeling and Simulation, Independent set, Maximal independent set, PARKING PROCESS, CONFIGURATION MODEL, CIENCIAS NATURALES Y EXACTAS, Mathematics - Probability
Abstract

By constructing jointly a random graph and an associated exploration process, we define the dynamics of a "parking process" on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald., Comment: keywords: random graphs, hydrodynamic limit, parking process

Published
2017
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11. Ergodic decompositions of stationary max-stable processes in terms of their spectral functions

Author

Zakhar Kabluchko, Clément Dombry, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Institut für Mathematische Stochastik, and Georg-August-University [Göttingen]

Subjects
Statistics and Probability, de Haan representation, Pure mathematics, Mixed moving maximum process, 01 natural sciences, 010104 statistics & probability, Mixing (mathematics), Positive/null decomposition, Ergodic theory, Mathematics - Dynamical Systems, 0101 mathematics, Mixing process, Conservative/dissipative decomposition, Ergodic process, Max-stable random process, Mathematics, Applied Mathematics, 010102 general mathematics, Ergodicity, Null (mathematics), 16. Peace & justice, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Flow (mathematics), Non-singular flow, Modeling and Simulation, Bounded function, 60G70 (Primary), 60G52, 60G60, 60G55, 60G10, 37A10, 37A25 (Secondary), Dissipative system, Mathematics - Probability
Abstract

We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to $0$ in the Ces\`{a}ro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative iff it converges to $0$. Surprisingly, for such processes a spectral function is integrable a.s. iff it converges to $0$ a.s. Based on these results, we provide new criteria for ergodicity, mixing, and existence of a mixed moving maximum representation of a stationary max-stable process in terms of its spectral functions. In particular, we study a decomposition of max-stable processes which characterizes the mixing property., Comment: 21 pages, no figures

Published
2017
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12. New deviation inequalities for martingales with bounded increments

Author

Emmanuel Rio, Laboratoire de Mathématiques de Versailles (LMV), and Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Discrete mathematics, Inequality, Applied Mathematics, media_common.quotation_subject, 010102 general mathematics, 01 natural sciences, Bennett's inequality, 010104 statistics & probability, Modeling and Simulation, Bounded function, Azuma's inequality, [MATH]Mathematics [math], 0101 mathematics, Random variable, ComputingMilieux_MISCELLANEOUS, Hoeffding's inequality, Mathematics, media_common
Abstract

In this paper we give new deviation inequalities for martingales with increments bounded from above. Our results are based on an improvement of the results of Bennett (1968) for random variables bounded from above.

Published
2017
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13. Spectral homogenization of reversible random walks on Zd in a random environment

Author

Jérôme Depauw, Daniel Boivin, Laboratoire de Mathématiques (LM-Brest), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et Physique Théorique (LMPT), Université de Tours-Centre National de la Recherche Scientifique (CNRS), Fédération de recherche Denis Poisson (FDP), Université d'Orléans (UO)-Université de Tours-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Tours, Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), and Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO)

Subjects
Statistics and Probability, Central limit theorem, 01 natural sciences, Homogenization (chemistry), 010104 statistics & probability, symbols.namesake, Dirichlet eigenvalue, 35B27, 35R60, 60K37, 82B41, Mathematics::Probability, Modelling and Simulation, Exit times, Difference operators, 0101 mathematics, Eigenvalues and eigenvectors, Mathematics, Dirichlet problem, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Hilbert space, Random walk, Random media, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Convergence of random variables, Modeling and Simulation, symbols, Dirichlet eigenvalues
Abstract

International audience; The first result is a homogenization theorem for the Dirichlet eigenvalues of reversible random walks on Z^d with stationary and uniformly elliptic conductances. It is then used to prove that the CLT holds in μ-almost all environments and to study the law of the exit times. Applications to the almost sure convergence of capacities and currents are given in the last section.

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14. Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment

Author

Quansheng Liu, Frédérique Watbled, Laboratoire de Mathématiques et Applications des Mathématiques, EA 3885 (LMAM), Université de Bretagne Sud (UBS), Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), and Université de Brest (UBO)-Université de Bretagne Sud (UBS)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Independent and identically distributed random variables, Statistics and Probability, Bernstein's inequality, Supermartingales, Large deviation inequality, super-martingales, Primary 60G42,60K35, secondary 60K37, 60G50, 60F10, 82D30, 01 natural sciences, Upper and lower bounds, Directed polymers, Combinatorics, 010104 statistics & probability, Martingale differences, Modelling and Simulation, Almost surely, 0101 mathematics, Concentration inequality, Free energy, Mathematics, Exponential inequality, Random environment, Applied Mathematics, 010102 general mathematics, Multiplicative function, 60K37, 60G50, 60F10, 82D30 (Secondary), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Convergence of random variables, Bernstein’s inequality, Modeling and Simulation, Convergence rate, 60G42,60K35 (Primary), Hoeffding-Azuma's inequality, Hoeffding and Azuma’s inequality, multiplicative cascades, Martingale (probability theory), Mathematics - Probability, Multiplicative cascades
Abstract

The objective of the present paper is to establish exponential large deviation inequalities, and to use them to show exponential concentration inequalities for the free energy of a polymer in general random environment, its rate of convergence, and an expression of its limit value in terms of those of some multiplicative cascades., Comment: 25 pages

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15. Marginal densities of the 'true' self-repelling motion

Author

Bálint Tóth, Laure Dumaz, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Limit of a function, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Motion (geometry), 01 natural sciences, 010104 statistics & probability, Scaling limit, Distribution (mathematics), Position (vector), Joint probability distribution, Modeling and Simulation, Local time, FOS: Mathematics, [MATH]Mathematics [math], 0101 mathematics, Marginal distribution, Mathematics - Probability, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

Let X(t) be the true self-repelling motion (TSRM) constructed by B.T. and Wendelin Werner in 1998, L(t,x) its occupation time density (local time) and H(t):=L(t,X(t)) the height of the local time profile at the actual position of the motion. The joint distribution of (X(t),H(t)) was identified by B.T. in 1995 in somewhat implicit terms. Now we give explicit formulas for the densities of the marginal distributions of X(t) and H(t). The distribution of X(t) has a particularly surprising shape: It has a sharp local minimum with discontinuous derivative at 0. As a consequence we also obtain a precise version of the large deviation estimate of arXiv:1105.2948v3., Comment: 20 pages, 7 figures

Published
2013
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16. Hitting times for the perturbed reflecting random walk

Author

Laurent Serlet, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, 01 natural sciences, k-nearest neighbors algorithm, 010104 statistics & probability, Recurrence, Modelling and Simulation, 0101 mathematics, Perturbed random walk, ComputingMilieux_MISCELLANEOUS, Brownian motion, Hitting times, Once reinforced random walk, Mathematics, Law of the iterated logarithm, Heterogeneous random walk in one dimension, Invariance principle, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Loop-erased random walk, Random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Modeling and Simulation, Perturbed Brownian motion
Abstract

International audience; We consider a nearest neighbor random walk on Z which is reflecting at 0 and perturbed when it reaches its maximum. We compute the law of the hitting times and derive many corollaries, especially in-variance principles with (rather) explicit descriptions of the asymptotic laws. We obtain also some results on the almost sure asymptotic behavior. As a by-product one can derive results on the reflecting Brownian motion perturbed at its maximum.

Published
2013
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17. Uniform concentration inequality for ergodic diffusion processes observed at discrete times

Author

L. Galtchouk, S. Pergamenshchikov, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), and RFFI-Grant 09-01-00172-a

Subjects
Statistics and Probability, Stationary ergodic process, 01 natural sciences, 010104 statistics & probability, Modelling and Simulation, FOS: Mathematics, Concentration inequality, Ergodic theory, 0101 mathematics, Diffusion (business), Mathematics, Pointwise, Tail distribution, Markov chains, Markov chain, Geometric ergodicity, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Ergodicity, Mathematical analysis, Upper exponential bound, Ergodic diffusion processes, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Discrete time and continuous time, Modeling and Simulation, Mathematics - Probability, 60F10
Abstract

In this paper a concentration inequality is proved for the deviation in the ergodic theorem for diffusion processes in the case of discrete time observations. The proof is based on geometric ergodicity of diffusion processes. We consider as an application the nonparametric pointwise estimation problem of the drift coefficient when the process is observed at discrete times.

Published
2013
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18. New techniques for empirical processes of dependent data

Author

Herold Dehling, Olivier Durieu, Dalibor Volny, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), and Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, 62G30, Pure mathematics, Stationary process, Chaining, Dynamical systems theory, Markov chain, Mathematics - Statistics Theory, Statistics Theory (math.ST), Dynamical system, 01 natural sciences, Combinatorics, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Transfer operator, Modelling and Simulation, FOS: Mathematics, 0101 mathematics, Empirical process, Mathematics, Central limit theorem, 60G10, 60F17, Invariance principle, Applied Mathematics, Probability (math.PR), 010102 general mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Modeling and Simulation, Mathematics - Probability
Abstract

We present a new technique for proving the empirical process invariance principle for stationary processes ( X n ) n ≥ 0 . The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions ( f ( X n ) ) n ≥ 0 , not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques.

Published
2009
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19. New large deviation results for some super-Brownian processes

Author

Laurent Serlet, Serlet, Laurent, Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Stochastic process, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Excursion, Motion (geometry), 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Mathematics::Probability, Extinction (optical mineralogy), Modeling and Simulation, Modelling and Simulation, Large deviations theory, 0101 mathematics, Representation (mathematics), Constant (mathematics), Brownian motion, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We give large deviation results for the super-Brownian excursion conditioned to have unit mass or unit extinction time and for super-Brownian motion with constant non-positive drift. We use a representation of these processes by a path-valued process, the so-called Brownian snake for which we state large deviation principles.

Published
2009
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20. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

Author

Bruno Saussereau, David Nualart, Department of mathematics, Kansas university, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Geometric Brownian motion, Fractional Brownian motion, Stochastic differential equation, Lebesgue measure, Differential equation, Applied Mathematics, Malliavin calculus, 010102 general mathematics, Mathematical analysis, Stochastic calculus, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Mathematics::Probability, Diffusion process, Modelling and Simulation, Modeling and Simulation, 0101 mathematics, Mathematics
Abstract

International audience; We prove the Malliavin regularity of the solution of a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H > 0:5. The result is based on the Fréchet differentiability with respect to the input function for deterministic differential equations driven by Hölder continuous functions. It is also shown that the law of the solution has a density with respect to the Lebesgue measure, under a suitable nondegeneracy condition

Published
2009
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21. Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths

Author

Philippe Carmona, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
Statistics and Probability, Lyapunov function, Differential equation, heat bath, 01 natural sciences, 010104 statistics & probability, symbols.namesake, harmonic oscillator, Modelling and Simulation, FOS: Mathematics, Uniqueness, 0101 mathematics, Hamiltonian chain, Mathematics, Mathematical physics, Probability measure, invariant measure, 60K35, 60J65, 82C99, Stochastic process, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Statistical mechanics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Modeling and Simulation, symbols, Invariant measure, Mathematics - Probability, Stationary state
Abstract

In this note we consider a chain of $N$ oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov's function technique of Rey-Bellet and coauthors with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by respectively Bernardin and Olla, Lefevere and Schenkel, Submitted

Published
2007
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22. Monotonicity properties of multi-dimensional reflected diffusions in random environment and applications

Author

Landy Rabehasaina, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)

Subjects
Statistics and Probability, Stationary process, Stochastic networks, Fork–join queue, 01 natural sciences, 010104 statistics & probability, Reflected stochastic differential equation, Modelling and Simulation, Computer Science::Networking and Internet Architecture, Fluid queue, Applied mathematics, Ergodic theory, 60K25, 60K30, 0101 mathematics, Queue, Ergodic process, Mathematics, Stochastic process, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Ergodicity, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Computer Science::Performance, Fluid queues, Modeling and Simulation
Abstract

We consider an N -dimensional reflected process, modeling an infinite capacity fluid queues network, of which service and input rates depend on the queue levels as well as on the state of an exterior ergodic stationary process. N is the number of queues in the network. We prove a monotonicity result for such a process, from which we deduce stability results for networks of queues. Particular attention is paid to the case N = 2 . Next, we give some applications of those stability results.

Published
2006
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23. Kahane–Khintchine inequalities and functional central limit theorem for stationary random fields

Author

Mohamed El Machkouri, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), and Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Pure mathematics, Metric entropy, Orlicz spaces, Functional central limit theorem, 01 natural sciences, 010104 statistics & probability, Modelling and Simulation, Mixing random fields, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Brownian motion, Central limit theorem, Mathematics, Martingale difference random fields, Random field, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Random function, Invariance principle, Exponential function, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Real-valued function, Modeling and Simulation, Bounded function, Martingale difference sequence, Kahane–Khintchine inequalities
Abstract

We establish new Kahane–Khintchine inequalities in Orlicz spaces induced by exponential Young functions for stationary real random fields which are bounded or satisfy some finite exponential moment condition. Next, we give sufficient conditions for partial sum processes indexed by classes of sets satisfying some metric entropy condition to converge in distribution to a set-indexed Brownian motion. Moreover, the class of random fields that we study includes φ-mixing and martingale difference random fields.

Published
2002
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24. Super-Brownian motion conditioned on the total mass

Author

Laurent Serlet, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), and Serlet, Laurent

Subjects
Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Super-Brownian motion, 01 natural sciences, Combinatorics, 010104 statistics & probability, Mathematics::Probability, Modelling and Simulation, Integrated super-Brownian excursion, 0101 mathematics, Martingale problem, Brownian motion, ComputingMilieux_MISCELLANEOUS, Mathematics, Particle system, Galton–Watson tree, Stochastic process, Applied Mathematics, 010102 general mathematics, Excursion, Mathematical analysis, Brownian excursion, Statistical mechanics, Galton–Watson process, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Modeling and Simulation, Martingale (probability theory)
Abstract

We study the super-Brownian motion ( X t ) conditioned on the total mass Z = ∫ 0 + ∞ X t ( 1 ) d t as the continuous limit of a system of branching trajectories which genealogical structure is a Galton–Watson tree conditioned on the number of its vertices. We characterize this process by a martingale problem and give a “snake” representation. Then we apply these results to a process that we call super-Brownian excursion. Its integral is the so-called Integrated Super-Brownian Excursion that has appeared recently as limit of several systems of statistical mechanics.

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25. Totally asymmetric attractive particle systems on Z hydrodynamic limit for general initial profiles

Author

Jean-Pierre Fouque, Ellen Saada, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), and Saada, Ellen

Subjects
Statistics and Probability, Attractiveness, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], media_common.quotation_subject, Integer lattice, Geometry, Law of large numbers, 01 natural sciences, Local equilibrium, Asymmetry, k-nearest neighbors algorithm, 010104 statistics & probability, Modelling and Simulation, Statistical physics, 0101 mathematics, Entropy (arrow of time), ComputingMilieux_MISCELLANEOUS, Mathematics, media_common, Particle system, Applied Mathematics, 010102 general mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Hydrodynamic equation, Modeling and Simulation, Coupling techniques, Entropy solution, Euler scaling
Abstract

We extend previous results on the preservation of local equilibrium for one dimensional totally asymmetric attractive particle systems, the simple exclusion process and the zero range process. The hydrodynamic behavior is studied for general initial profiles when particles are jumping only to the nearest neighbor in a given direction.

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26. A Markov additive risk process in dimension 2 perturbed by a fractional Brownian motion

Author

Landy Rabehasaina, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)

Subjects
Hurst exponent, Reinsurance, Statistics and Probability, Mathematical optimization, Fractional Brownian motion, Markov chain, Markov additive process, Applied Mathematics, 010102 general mathematics, Bivariate analysis, Ruin theory, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Modeling and Simulation, Modelling and Simulation, Applied mathematics, 0101 mathematics, First-hitting-time model, ComputingMilieux_MISCELLANEOUS, Multivariate risk theory, Mathematics, First time passage process
Abstract

We consider the following theoretical reinsurance ruin problem. An insurance company has two types of independent claims, respectively modeled by a Markov additive process (large claims) and a fractional Brownian motion (small claims) with Hurst parameter H ∈ [ 1 / 2 , 1 ) , and chooses to reinsure both of them according to a quota share policy. This leads to studying a bivariate risk process. We study two types of ruins, corresponding to either ruin of one of the risk processes, or of both. We obtain asymptotics of the corresponding ruin probabilities when initial reserves tend to infinity along a direction.

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27. Boundary homogenization in domains with randomly oscillating boundary

Author

Olivier Bodart, Youcef Amirat, Andrey Piatnitski, Gregory A. Chechkin, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Faculty of Mechanics and Mathematics (Mech-Mat), Moscow State Lomonossov University, Narvik University College (HiN), Narvik University College, P. N. Lebedev Physical Institute of the Russian Academy of Sciences [Moscow] (LPI RAS), Russian Academy of Sciences [Moscow] (RAS), and Université d'État Lomonossov de Moscou = Lomonosov Moscow State University (MSU)

Subjects
Statistics and Probability, Random oscillating boundary, 01 natural sciences, Boundary homogenization, symbols.namesake, Modelling and Simulation, Free boundary problem, Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mixed boundary condition, 16. Peace & justice, Singular boundary method, Robin boundary condition, 010101 applied mathematics, Modeling and Simulation, Dirichlet boundary condition, Effective boundary condition, symbols, Cauchy boundary condition
Abstract

International audience; We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface A Fourier boundary condition with random coefficients is imposed on the oscillating boundary We derive the effective boundary condition prove a convergence result and establish error estimates

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