468 results on '"60G55"'
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2. There is no stationary cyclically monotone Poisson matching in 2d.
- Author
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Huesmann, Martin, Mattesini, Francesco, and Otto, Felix
- Subjects
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POISSON processes , *MATHEMATICS , *MARTINGALES (Mathematics) - Abstract
We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension d = 2 . The proof combines the harmonic approximation result from Goldman et al. (Commun. Pure Appl. Math. 74:2483–2560, 2021) with local asymptotics for the two-dimensional matching problem for which we give a new self-contained proof using martingale arguments. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations.
- Author
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Scardia, Lucia, Zemas, Konstantinos, and Zeppieri, Caterina Ida
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DIRICHLET problem , *NONLINEAR equations , *RANDOM sets , *MATHEMATICS , *EQUATIONS - Abstract
In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^n$$\end{document}. Under the assumption that the perforations are small balls whose centres and radii are generated by a
stationary short-range marked point process , we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite (n-q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-q)$$\end{document}-moment, where 1q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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4. Applications of a change of measures technique for compound mixed renewal processes to the ruin problem
- Author
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Spyridon M. Tzaninis
- Subjects
Compound mixed renewal process ,change of measures ,progressively equivalent measures ,regular conditional probabilities ,ruin probability ,60G55 ,Applied mathematics. Quantitative methods ,T57-57.97 ,Mathematics ,QA1-939 - Abstract
In the present paper the change of measures technique for compound mixed renewal processes, developed in Tzaninis and Macheras [ArXiv:2007.05289 (2020) 1–25], is applied to the ruin problem in order to obtain an explicit formula for the probability of ruin in a mixed renewal risk model and to find upper and lower bounds for it.
- Published
- 2021
- Full Text
- View/download PDF
5. Spectral Rigidity of Random Schrödinger Operators via Feynman–Kac Formulas.
- Author
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Gaudreau Lamarre, Pierre Yves, Ghosal, Promit, and Liao, Yuchen
- Subjects
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SCHRODINGER operator , *RANDOM operators , *RANDOM noise theory , *MATHEMATICS - Abstract
We develop a technique for proving number rigidity (in the sense of Ghosh and Peres in Duke Math J 166(10):1789–1858, 2017) of the spectrum of general random Schrödinger operators (RSOs). Our method makes use of Feynman–Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman–Kac formulas for RSOs with multiplicative noise (Gaudreau Lamarre in Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise, Preprint arXiv:1902.05047v3, 2019; Gaudreau Lamarre and Shkolnikov in Ann Inst Henri Poincaré Probab Stat 55(3):1402–1438, 2019; Gorin and Shkolnikov in Ann Probab 46(4):2287–2344, 2018) by Gorin, Shkolnikov, and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form - 1 2 Δ + V + ξ , where V is a deterministic potential and ξ is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential V, and the singularity of the noise ξ . [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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6. Compound Poisson point processes, concentration and oracle inequalities.
- Author
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Zhang, Huiming and Wu, Xiaoxu
- Subjects
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POINT processes , *POISSON processes , *RANDOM variables , *RANDOM measures , *MATHEMATICAL equivalence , *MATHEMATICS , *BINOMIAL theorem - Abstract
This note aims at presenting several new theoretical results for the compound Poisson point process, which follows the work of Zhang et al. (Insur. Math. Econ. 59:325–336, 2014). The first part provides a new characterization for a discrete compound Poisson point process (proposed by Aczél (Acta Math. Hung. 3(3):219–224, 1952)), it extends the characterization of the Poisson point process given by Copeland and Regan (Ann. Math. 37:357–362, 1936). Next, we derive some concentration inequalities for discrete compound Poisson point process (negative binomial random variable with unknown dispersion is a significant example). These concentration inequalities are potentially useful in count data regression. We give an application in the weighted Lasso penalized negative binomial regressions whose KKT conditions of penalized likelihood hold with high probability and then we derive non-asymptotic oracle inequalities for a weighted Lasso estimator. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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7. Limites en champ moyen de processus de Hawkes en interaction, en régime diffusif
- Author
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Dasha Loukianova, Eva Löcherbach, Xavier Erny, Laboratoire de Mathématiques et Modélisation d'Evry (LaMME), Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Centre National de la Recherche Scientifique (CNRS), and Université Panthéon-Sorbonne (UP1)
- Subjects
Statistics and Probability ,60K35, 60G55, 60J35 ,Multivariate nonlinear Hawkes processes ,Poisson distribution ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Stochastic differential equation ,FOS: Mathematics ,Statistical physics ,Limit (mathematics) ,Piecewise deterministic Markov processes ,0101 mathematics ,Mathematics ,Sequence ,Mean field interaction ,Probability (math.PR) ,010102 general mathematics ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,MSC 2010 : 60K35 ,60G55 ,60J35 ,Distribution (mathematics) ,Convergence of random variables ,Mean field theory ,Limit point ,symbols ,Mathematics - Probability - Abstract
International audience; We consider a sequence of systems of Hawkes processes having mean field interactions in a diffusive regime. The stochastic intensity of each process is a solution of a stochastic differential equation driven by N independent Poisson random measures. We show that, as the number of interacting components N tends to infinity, this intensity converges in distribution in Skorohod space to a CIR-type diffusion. Moreover, we prove the convergence in distribution of the Hawkes processes to the limit point process having the limit diffusion as intensity. To prove the convergence results, we use analytical technics based on the convergence of the associated infinitesimal generators and Markovian semigroups.; Nous considérons des systèmes de processus de Hawkes, avec des interactions de type champ moyen, dans un régime diffusif. L'intensité stochastique de chaque processus est solution d'une équation différentielle stochastique dirigée par N mesures aléatoires de Poisson indépendantes. Nous montrons que, quand le nombre N de composants qui interagissent tend vers l'infini, cette intensité converge en loi dans l'espace de Skorokhod vers une diffusion de type CIR. De plus, nous démontrons la convergence en loi du système des processus de Hawkes vers un processus ponctuel ayant comme intensité stochastique la diffusion limite. Pour prouver cette convergence, nous utilisons des techniques analytiques basées sur la convergence des générateurs infinitésimaux et des semi-groupes associés à des processus de Markov.
- Published
- 2022
8. On the sum of independent generalized Mittag–Leffler random variables and the related fractional processes
- Author
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Fabrizio Cinque
- Subjects
Statistics and Probability ,alternating process ,60G50 ,Distribution (number theory) ,Mathematics::Complex Variables ,Applied Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,60G55 ,caputo derivative ,Generalized Mittag–Leffler distribution ,multivariate Mittag–Leffler function ,Primary 60G22 ,state-dependent process ,01 natural sciences ,010104 statistics & probability ,Mathematics::Probability ,Applied mathematics ,Point (geometry) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
We obtain the distribution of the sum of independent and non-identically distributed generalized Mittag–Leffler random variables. We then apply this result to study some related fractional point pr...
- Published
- 2022
9. Continuum Line-of-Sight Percolation on Poisson-Voronoi Tessellations
- Author
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Elie Cali, Taoufik En-Najjary, Bartlomiej Blaszczyszyn, Quentin Le Gall, Orange Labs [Chatillon], Orange Labs, Dynamics of Geometric Networks (DYOGENE), Département d'informatique - ENS Paris (DI-ENS), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria), Eurecom [Sophia Antipolis], Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Département d'informatique - ENS Paris (DI-ENS), Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-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), Département d'informatique de l'École normale supérieure (DI-ENS), École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris)
- Subjects
Statistics and Probability ,Poisson-Voronoi tessellation ,01 natural sciences ,Point process ,Cox process ,010104 statistics & probability ,[INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI] ,FOS: Mathematics ,Statistical physics ,0101 mathematics ,Mathematics ,Primary : 60K35, 60G55, 60D05, Secondary : 68M10, 90B15 ,Random graph ,Tessellation ,Continuum (topology) ,Euclidean space ,Applied Mathematics ,010102 general mathematics ,Probability (math.PR) ,Renormalisation ,Percolation ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,2010 Mathematics Subject Classification: Primary 60K35 ,60G55 ,60D05Secondary 68M10 ,90B15 ,Voronoi diagram ,Mathematics - Probability ,Simulation - Abstract
In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson-Voronoi tessellation in the $d$-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition $Z$ of these two processes, two points of $Z$ are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0-1 law, a subcritical phase as well as a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation., 30 pages, 4 figures. Accepted for publication in Advances in Applied Probability
- Published
- 2021
10. Local laws and rigidity for Coulomb gases at any temperature
- Author
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Sylvia Serfaty and Scott N. Armstrong
- Subjects
Statistics and Probability ,Surface (mathematics) ,Superadditivity ,FOS: Physical sciences ,Rigidity (psychology) ,01 natural sciences ,Microscopic scale ,Point process ,010104 statistics & probability ,symbols.namesake ,Subadditivity ,FOS: Mathematics ,Coulomb ,0101 mathematics ,Gibbs measure ,Mathematical Physics ,point process ,Mathematics ,large deviations principle ,Coulomb gas ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,49S05 ,82B05, 60G55, 60F10, 49S05 ,rigidity ,Law ,symbols ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,82B05 ,60F10 - Abstract
We study Coulomb gases in any dimension $d \geq 2$ and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exhibit a new minimal lengthscale for rigidity depending on the temperature. They apply as well to energy minimizers (formally zero temperature). The method is based on a bootstrap on scales and reveals the additivity of the energy modulo surface terms, via the introduction of subadditive and superadditive approximate energies., Comment: 87 pages, a computational mistake in the proof of Prop. 4.5 corrected. Changes compared to the published version in Annals of Probability are highlighted in color
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- 2021
11. Random moments for the new eigenfunctions of point scatterers on rectangular flat tori
- Author
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Henrik Ueberschär, Thomas Letendre, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0011,SpInQS,Géométrie spectrale de systèmes quantiques intermédiaires(2017), ANR-17-CE40-0008,UNIRANDOM,Universalité pour les domaines nodaux aléatoires(2017), and Centre National de la Recherche Scientifique (CNRS)
- Subjects
Nuclear and High Energy Physics ,Random wave model Mathematics Subject Classification 2010: 35P20 ,Distribution (number theory) ,Gaussian ,FOS: Physical sciences ,58J50 ,01 natural sciences ,Mathematics - Spectral Theory ,symbols.namesake ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,0103 physical sciences ,Poisson point process ,FOS: Mathematics ,Limit (mathematics) ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematical Physics ,Mathematics ,Laplace transform ,010308 nuclear & particles physics ,Random wave model ,Diophantine equation ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Statistical and Nonlinear Physics ,Torus ,Mathematical Physics (math-ph) ,Eigenfunction ,81Q50 ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Point scatterer ,symbols ,MSC2010: 35P20, 58J50, 60G55, 81Q50 ,60G55 ,Mathematics - Probability ,Quantum chaos ,[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] - Abstract
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 $$+\infty $$ .
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- 2021
12. Bootstrap estimators for the tail-index and for the count statistics of graphex processes
- Author
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Daniel M. Roy, Victor Veitch, Zacharie Naulet, and Ekansh Sharma
- Subjects
Statistics and Probability ,Random graph ,count statistics ,estimation ,60G70 ,Nonparametric statistics ,Probabilistic logic ,Estimator ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Graphex processes ,tail-index ,Primary 62F10, secondary 60G55, 60G70 ,sparse random graphs ,Econometrics ,FOS: Mathematics ,Point (geometry) ,60G55 ,Statistics, Probability and Uncertainty ,bootstrap ,Tail index ,62F10 ,Mathematics ,Parametric statistics - Abstract
Graphex processes resolve some pathologies in traditional random graph models, notably, providing models that are both projective and allow sparsity. Most of the literature on graphex processes study them from a probabilistic point of view. Techniques for inferring the parameter of these processes -- the so-called \textit{graphon} -- are still marginal; exceptions are a few papers considering parametric families of graphons. Nonparametric estimation remains unconsidered. In this paper, we propose estimators for a selected choice of functionals of the graphon. Our estimators originate from the subsampling theory for graphex processes, hence can be seen as a form of bootstrap procedure.
- Published
- 2021
13. Functional large deviations for Cox processes and $\mathit{Cox}/G/\infty$ queues, with a biological application
- Author
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Ayalvadi Ganesh, Edward Crane, and Justin Dean
- Subjects
Statistics and Probability ,Gene regulatory network ,random measures ,01 natural sciences ,Point process ,Cox process ,010104 statistics & probability ,60K25 ,Applied mathematics ,0101 mathematics ,Queue ,point processes ,Mathematics ,General distribution ,infinite-server queues ,010102 general mathematics ,Process (computing) ,chemical reaction networks ,Service process ,Computer Science::Performance ,Large deviations ,60G57 ,Large deviations theory ,60G55 ,Statistics, Probability and Uncertainty ,60F10 - Abstract
We consider an infinite-server queue into which customers arrive according to a Cox process and have independent service times with a general distribution. We prove a functional large deviations principle for the equilibrium queue length process. The model is motivated by a linear feed-forward gene regulatory network, in which the rate of protein synthesis is modulated by the number of RNA molecules present in a cell. The system can be modelled as a nonstandard tandem of infinite-server queues, in which the number of customers present in a queue modulates the arrival rate into the next queue in the tandem. We establish large deviation principles for this queueing system in the asymptotic regime in which the arrival process is sped up, while the service process is not scaled.
- Published
- 2020
14. Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery
- Author
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Markus Reiss, Anselm Johannes Schmidt-Hieber, and Mathematics of Operations Research
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Statistics and Probability ,62C10 ,Boundary (topology) ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,posterior contraction ,Poisson distribution ,compound Poisson process ,symbols.namesake ,Frequentist inference ,Compound Poisson process ,Poisson point process ,Prior probability ,FOS: Mathematics ,Applied mathematics ,62G05 ,62C10, 62G05, 60G55 ,Bernstein–von Mises theorem ,Mathematics ,subordinator prior ,boundary detection ,Function (mathematics) ,Frequentist Bayes analysis ,symbols ,60G55 ,Statistics, Probability and Uncertainty - Abstract
Given data from a Poisson point process with intensity $(x,y) \mapsto n \mathbf{1}(f(x)\leq y),$ frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone and piecewise constant support boundaries and adapts to H\"older smooth boundaries with smoothness index at most one. We then derive a non-standard Bernstein-von Mises result for a compound Poisson process prior and a function space with increasing parameter dimension. As an intermediate result the limiting shape of the posterior for random histogram type priors is obtained. In both settings, it is shown that the marginal posterior of the functional $\vartheta =\int f$ performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, $(1-\alpha)$-credible sets are also asymptotic $(1-\alpha)$-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions indicating that credible sets only have frequentist coverage for priors that are specifically constructed to match properties of the underlying true function., Comment: The first version of arXiv:1703.08358 has been expanded and rewritten. We decided to split it in two separate papers, a new version of arXiv:1703.08358 and this article
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- 2020
15. Stability and mean-field limits of age dependent Hawkes processes
- Author
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Eva Löcherbach, Susanne Ditlevsen, and Mads Bonde Raad
- Subjects
Age dependency ,Statistics and Probability ,Multivariate point processes ,Age dependent ,Multivariate nonlinear Hawkes processes ,General Relativity and Quantum Cosmology ,Coupling ,60K05 ,60G57 ,60G55 ,Piecewise deterministic Markov processes ,Statistics, Probability and Uncertainty ,Mean-field approximations ,Stability ,Humanities ,Mathematics - Abstract
Depuis la derniere decennie il s’est avere que la classe des processus de Hawkes fournit un bon modele pour decrire la connectivite fonctionnelle dans un reseau de neurones. Dans cet article nous etudions une variante de ce processus, le processus de Hawkes structure en âge. Cette structure en âge rajoute un comportement individuel apres les sauts a la dynamique de chaque composante, ce qui permet en particulier de decrire une periode refractaire durant laquelle l’influence du reseau est supprimee ou au moins modifiee. Nous ameliorons les resultats de stabilite classiques pour les processus de Hawkes dans ce cadre. En particulier, nous n’avons ni besoin de supposer que les intensites sont bornees, ni d’imposer une condition aux normes Lipschitz des fonctions taux de saut. Lorsque les interactions entre les neurones sont du type champ moyen, nous etudions les limites en grande population et nous demontrons la propriete de propagation du chaos du systeme.
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- 2020
16. Random walks on dynamical random environments with nonuniform mixing
- Author
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Marcelo R. Hilario, Augusto Teixeira, and Oriane Blondel
- Subjects
Statistics and Probability ,strong law of large numbers ,82B43 ,010102 general mathematics ,Markov process ,Invariant (physics) ,Random walk ,01 natural sciences ,renormalization ,010104 statistics & probability ,symbols.namesake ,Law of large numbers ,60K35 ,symbols ,Exponent ,dynamic random environments ,Spectral gap ,Statistical physics ,60G55 ,0101 mathematics ,Statistics, Probability and Uncertainty ,Concentration inequality ,Random walks ,Mixing (physics) ,Mathematics - Abstract
In this paper, we study random walks on dynamical random environments in $1+1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a concentration inequality around the asymptotic speed. The mixing hypothesis imposes a polynomial decay rate of covariances on the environment with sufficiently high exponent but does not impose uniform mixing. Examples of environments for which our methods apply include the contact process and Markovian environments with a positive spectral gap, such as the East model. For the East model, we also obtain that the distinguished zero satisfies a law of large numbers with strictly positive speed.
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- 2020
17. Finitary isomorphisms of Brownian motions
- Author
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Zemer Kosloff and Terry Soo
- Subjects
Statistics and Probability ,37A35 ,Rational number ,Pure mathematics ,renewal point processes ,Dynamical Systems (math.DS) ,finitary isomorphisms ,Bernoulli's principle ,Mathematics::Probability ,reflected Brownian motions ,FOS: Mathematics ,Finitary ,Almost surely ,Mathematics - Dynamical Systems ,Brownian motion ,Mathematics ,37A35, 60G15, 60G55, 60J10 ,Probability (math.PR) ,Ornstein theory ,Flow (mathematics) ,Bounded function ,60G15 ,60J10 ,Isomorphism ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
Ornstein and Shields (Advances in Math., 10:143-146, 1973) proved that Brownian motion reflected on a bounded region is an infinite entropy Bernoulli flow and thus Ornstein theory yielded the existence of a measure-preserving isomorphism between any two such Brownian motions. For fixed h >0, we construct by elementary methods, isomorphisms with almost surely finite coding windows between Brownian motions reflected on the intervals [0, qh] for all positive rationals q., Published at https://doi.org/10.1214/19-AOP1412 in the Annals of Probability by the Institute of Mathematical Statistics
- Published
- 2020
18. Ergodic Poisson splittings
- Author
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Emmanuel Roy, Thierry de la Rue, Elise Janvresse, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA), Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), 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 GdR GeoSto
- Subjects
Statistics and Probability ,Poisson suspension ,Pure mathematics ,splitting ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Dynamical Systems (math.DS) ,Poisson distribution ,01 natural sciences ,Point process ,010104 statistics & probability ,symbols.namesake ,Poisson point process ,FOS: Mathematics ,Ergodic theory ,37A50 ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics ,Probability (math.PR) ,010102 general mathematics ,thinning ,37A40 ,joinings ,Invariant (physics) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Random measure ,Transformation (function) ,symbols ,Equivariant map ,60G57 ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,random measure - Abstract
International audience; In this paper we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further application is derived, giving a full description of the structure of a random measure invariant under the action of the transformation.
- Published
- 2020
19. Stable matchings in high dimensions via the Poisson-weighted infinite tree
- Author
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Alexander E. Holroyd, Yuval Peres, and James B. Martin
- Subjects
Statistics and Probability ,05C70 ,Probability (math.PR) ,Poisson process ,Astrophysics::Cosmology and Extragalactic Astrophysics ,Poisson distribution ,60D05, 60G55, 05C70 ,Combinatorics ,symbols.namesake ,Tree (descriptive set theory) ,Poisson-weighted infinite tree ,Stable matching ,symbols ,FOS: Mathematics ,Astrophysics::Solar and Stellar Astrophysics ,60G55 ,Statistics, Probability and Uncertainty ,60D05 ,Mathematics - Probability ,Point process ,Mathematics - Abstract
We consider the stable matching of two independent Poisson processes in $\mathbb{R}^d$ under an asymmetric color restriction. Blue points can only match to red points, while red points can match to points of either color. It is unknown whether there exists a choice of intensities of the red and blue processes under which all points are matched. We prove that for any fixed intensities, there are unmatched blue points in sufficiently high dimension. Indeed, if the ratio of red to blue intensities is $\rho$ then the intensity of unmatched blue points converges to $e^{-\rho}/(1+\rho)$ as $d\to\infty$. We also establish analogous results for certain multi-color variants. Our proof uses stable matching on the Poisson-weighted infinite tree (PWIT), which can be analyzed via differential equations. The PWIT has been used in many settings as a scaling limit for models involving complete graphs with independent edge weights, but we believe that this is the first rigorous application to high-dimensional Euclidean space. Finally, we analyze the asymmetric matching problem under a hierarchical metric, and show that there are unmatched points for all intensities., Comment: 29 pages
- Published
- 2020
20. The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise
- Author
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Péter Kevei and Carsten Chong
- Subjects
Statistics and Probability ,strong law of large numbers ,Additive intermittency ,Limit superior and limit inferior ,stochastic heat equation ,Noise (electronics) ,levy noise ,Law of large numbers ,intermittency ,Poisson noise ,stochastic PDE ,Mathematics ,Sequence ,superpositions ,integral test ,35B40 ,Mathematical analysis ,Shot noise ,almost-sure asymptotics ,White noise ,Lévy noise ,60G17 ,Bounded function ,60H15 ,Heat equation ,60F15 ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
We examine the almost-sure asymptotics of the solution to the stochastic heat equation driven by a L\'evy space-time white noise. When a spatial point is fixed and time tends to infinity, we show that the solution develops unusually high peaks over short time intervals, even in the case of additive noise, which leads to a breakdown of an intuitively expected strong law of large numbers. More precisely, if we normalize the solution by an increasing nonnegative function, we either obtain convergence to $0$, or the limit superior and/or inferior will be infinite. A detailed analysis of the jumps further reveals that the strong law of large numbers can be recovered on discrete sequences of time points increasing to infinity. This leads to a necessary and sufficient condition that depends on the L\'evy measure of the noise and the growth and concentration properties of the sequence at the same time. Finally, we show that our results generalize to the stochastic heat equation with a multiplicative nonlinearity that is bounded away from zero and infinity., Comment: Forthcoming in The Annals of Probability
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- 2020
21. Brownian motion in attenuated or renormalized inverse-square Poisson potential
- Author
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Peter Nelson and Renato Soares dos Santos
- Subjects
Statistics and Probability ,Brownian motion in Poisson potential ,Probability (math.PR) ,Inverse square potential ,Inverse ,35J10 ,Poisson distribution ,60J65, 60G55, 60K37, 35J10, 35P15 ,Square (algebra) ,symbols.namesake ,60K37 ,Multipolar Hardy inequality ,FOS: Mathematics ,symbols ,60J65 ,Parabolic Anderson model ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Brownian motion ,Mathematics ,Mathematical physics ,35P15 - Abstract
We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in $\mathbb{R}^d$, $d \geq 3$. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel $\mathfrak{K}$ behaving as $\mathfrak{K}(x) \approx \theta |x|^{-2}$ near the origin, where $\theta \in (0,(d-2)^2/16]$. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that $\mathfrak{K}$ is integrable at infinity) or, when $d=3$, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in $d=3$ the problem with critical parameter $\theta = 1/16$, left open by Chen and Rosinski in [arXiv:1103.5717]., Comment: 36 pages
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- 2020
22. Donsker’s theorem in Wasserstein-1 distance
- Author
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Laurent Decreusefond and Laure Coutin
- Subjects
Statistics and Probability ,Donsker theorem ,Malliavin calculus ,010102 general mathematics ,Mathematical analysis ,Stein’s method ,Stein's method ,Lipschitz continuity ,Random walk ,01 natural sciences ,010104 statistics & probability ,60H07 ,Mathematics::Probability ,Rate of convergence ,Local time ,60G15 ,Wasserstein distance ,60F15 ,60G55 ,0101 mathematics ,Statistics, Probability and Uncertainty ,Donsker's theorem ,Brownian motion ,Mathematics - Abstract
We compute the Wassertein-1 (or Kantorovitch-Rubinstein) distance between a random walk in $\mathbf{R} ^{d}$ and the Brownian motion. The proof is based on a new estimate of the modulus of continuity of the solution of the Stein’s equation. As an application, we can evaluate the rate of convergence towards the local time at 0 of the Brownian motion and to a Brownian bridge.
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- 2020
23. Diffusions on a space of interval partitions: construction from marked Lévy processes
- Author
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Soumik Pal, Matthias Winkel, Noah Forman, and Douglas Rizzolo
- Subjects
Statistics and Probability ,Diffusion (acoustics) ,excursion theory ,branching processes ,interval partition ,Space (mathematics) ,01 natural sciences ,Lévy process ,Combinatorics ,Branching (linguistics) ,010104 statistics & probability ,60J25 ,Partition (number theory) ,0101 mathematics ,Mathematics ,60J60 ,self-similar diffusion ,60J80 ,Markov chain ,infinitely-many-neutral-alleles model ,010102 general mathematics ,Aldous diffusion ,Birth–death process ,60G18 ,Interval (graph theory) ,60G55 ,Statistics, Probability and Uncertainty ,Ray-Knight theorem ,60G52 - Abstract
Consider a spectrally positive Stable$(1\!+\!\alpha )$ process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning “sizes” varying during the lifetime. As for Crump–Mode–Jagers processes (with “characteristics”), we consider for each level the collection of individuals alive. We arrange their “sizes” at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable$(1\!+\!\alpha )$ process, this yields new theorems of Ray–Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson–Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.
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- 2020
24. No repulsion between critical points for planar Gaussian random fields
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Igor Wigman, Valentina Cammarota, and Dmitry Beliaev
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Statistics and Probability ,Field (physics) ,critical points ,Gaussian ,Diagonal ,FOS: Physical sciences ,01 natural sciences ,Point process ,010104 statistics & probability ,symbols.namesake ,Correlation function ,gaussian fields ,58K05 ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,Mathematics ,60G60 ,Random field ,Probability (math.PR) ,010102 general mathematics ,Isotropy ,Mathematical analysis ,Mathematical Physics (math-ph) ,Gaussian fields ,symbols ,60G55 ,Statistics, Probability and Uncertainty ,Asymptotic expansion ,Mathematics - Probability - Abstract
We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a 'generic' field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index., This version of the paper contains the text of the paper accepted for publication in 'Electronic Communications in Probability' and three appendices which show some technical computations. arXiv admin note: text overlap with arXiv:1704.04943
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- 2020
25. How long is the convex minorant of a one-dimensional random walk?
- Author
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Zakhar Kabluchko, Gerold Alsmeyer, Alexander Marynych, and Vladislav Vysotsky
- Subjects
Statistics and Probability ,Discrete mathematics ,Independent identically distributed ,60F05, 60G55 (Primary), 60J10 (Secondary) ,Probability (math.PR) ,random permutation ,Regular polygon ,Random permutation ,convex minorant ,Random walk ,random walk ,60F05 ,FOS: Mathematics ,60J10 ,60G55 ,Limit (mathematics) ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Mathematics - Probability ,Mathematics - Abstract
We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized., 21 pages, 1 figure, to appear in the Electronic Journal of Probability
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- 2020
26. Stability for Hawkes processes with inhibition
- Author
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Mads Bonde Raad and Eva Löcherbach
- Subjects
Statistics and Probability ,Lyapunov function ,education.field_of_study ,Weight function ,Spectral radius ,Population ,Probability (math.PR) ,stability ,Stability (probability) ,60G55, 60G57, 60J25, 60Fxx ,multivariate nonlinear Hawkes processes ,symbols.namesake ,60J25 ,symbols ,FOS: Mathematics ,60G57 ,60G55 ,Statistics, Probability and Uncertainty ,Biological system ,education ,piecewise deterministic Markov processes ,60Fxx ,Mathematics - Probability ,Mathematics ,Lyapunov functions - Abstract
We consider a multivariate non-linear Hawkes process in a multi-class setup where particles are organised within two populations of possibly different sizes, such that one of the populations acts excitatory on the system while the other population acts inhibitory on the system. The goal of this note is to present a class of Hawkes Processes with stable dynamics without assumptions on the spectral radius of the associated weight function matrix. This illustrates how inhibition in a Hawkes system significantly affects the stability properties of the system.
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- 2020
27. Diffusions on a space of interval partitions: construction from Bertoin’s ${\tt BES}_{0}(d)$, $d\in (0,1)$
- Author
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Winkel, M
- Subjects
Statistics and Probability ,60J80 ,Bessel process ,measure-valued diffusion ,Poisson–Dirichlet distribution ,excursion theory ,Markov process ,interval partition ,Space (mathematics) ,Lévy process ,Combinatorics ,symbols.namesake ,Dimension (vector space) ,60J25 ,60G18 ,symbols ,Partition (number theory) ,Interval (graph theory) ,60G55 ,Statistics, Probability and Uncertainty ,Bessel function ,Mathematics ,60J60 - Abstract
In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable L\'evy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.
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- 2020
28. On intermediate level sets of two-dimensional discrete Gaussian free field
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Oren Louidor and Marek Biskup
- Subjects
Statistics and Probability ,62G30 ,60G70 ,82B41 ,FOS: Physical sciences ,Gaussian Free Field ,Intermediate level ,01 natural sciences ,Combinatorics ,0103 physical sciences ,Gaussian free field ,FOS: Mathematics ,Scaling limit ,0101 mathematics ,Conformal invariance ,Mathematical Physics ,60G15, 82B41, 60G70, 62G30, 60G55, 60G57 ,Point process ,Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,Level set ,Liouville Quantum Gravity ,60G15 ,60G57 ,60G55 ,010307 mathematical physics ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattice) versions of suitably regular continuum domains $D\subset\mathbb C$ and describe the scaling limit, including local structure, of the level sets at heights growing as a $\lambda$-multiple of the height of the absolute maximum, for any $\lambda\in(0,1)$. We prove that, in the scaling limit, the scaled spatial position of a typical point $x$ sampled from this level set is distributed according to a Liouville Quantum Gravity (LQG) measure in $D$ at parameter equal $\lambda$-times its critical value, the field value at $x$ has an exponential intensity measure and the configuration near $x$ reduced by the value at $x$ has the law of a pinned DGFF reduced by a suitable multiple of the potential kernel. In particular, the law of the total size of the level set, properly-normalized, converges that that of the total mass of the LQG measure. This sharpens considerably an earlier conclusion by Daviaud., Comment: 41 pages, 3 figs
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- 2019
29. Mathematical Models of Gene Expression
- Author
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Philippe Robert, Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Robert, Philippe
- Subjects
Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Stochastic modelling ,Molecular Networks (q-bio.MN) ,Parameter Estimation ,marked Poisson point processes ,Context (language use) ,[SDV.BBM.BM] Life Sciences [q-bio]/Biochemistry, Molecular Biology/Molecular biology ,01 natural sciences ,Quantitative Biology - Quantitative Methods ,Quantitative Biology::Subcellular Processes ,010104 statistics & probability ,Convergence (routing) ,FOS: Mathematics ,State space ,Quantitative Biology - Molecular Networks ,Boolean Models ,Gene Regulation ,Statistical physics ,0101 mathematics ,Representation (mathematics) ,Quantitative Methods (q-bio.QM) ,Model Selection ,Mathematics ,Mathematical model ,Stochastic process ,Quantitative Biology::Molecular Networks ,010102 general mathematics ,Probability (math.PR) ,Thermodynamic Models ,[SDV.BBM.BM]Life Sciences [q-bio]/Biochemistry, Molecular Biology/Molecular biology ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Stochastic models ,Distribution (mathematics) ,Sensitivity Analysis ,Drosophila melanogaster ,92C40 ,FOS: Biological sciences ,8. Economic growth ,gene expression ,60G55 ,Differential Equation Models ,Mathematics - Probability - Abstract
International audience; In this paper we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, {\em the production process of proteins}. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.
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- 2019
30. The hyperbolic-type point process
- Author
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Nizar Demni, Pierre Lazag, 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), 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 ), 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
Pure mathematics ,Distribution (number theory) ,General Mathematics ,Weighted Bergman kernel ,Type (model theory) ,01 natural sciences ,Point process ,Determinantal point process ,Simple (abstract algebra) ,Ginibre-type point process ,0103 physical sciences ,Euclidean geometry ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Bergman kernel ,Probability (math.PR) ,010102 general mathematics ,Order (ring theory) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,60G55 ,010307 mathematical physics ,[ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR] ,46E22 ,Mathematics - Probability ,Poincaré disc - Abstract
In this paper, we introduce a two-parameters determinantal point process in the Poincar\'e disc and compute the asymptotics of the variance of its number of particles inside a disc centered at the origin and of radius $r$ as $r$ tends to $1$. Our computations rely on simple geometrical arguments whose analogues in the Euclidean setting provide a shorter proof of Shirai's result for the Ginibre-type point process. In the special instance corresponding to the weighted Bergman kernel, we mimic the computations of Peres and Virag in order to describe the distribution of the number of particles inside the disc., Comment: 12 pages
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- 2019
31. Finitary isomorphisms of Poisson point processes
- Author
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Terry Soo and Amanda Wilkens
- Subjects
37A35 ,Statistics and Probability ,Pure mathematics ,Probability (math.PR) ,finitary isomorphisms ,Poisson distribution ,Point process ,symbols.namesake ,General theory ,Poisson point process ,FOS: Mathematics ,symbols ,Finitary ,60G55 ,37A35, 60G10, 60G55 ,Isomorphism ,Statistics, Probability and Uncertainty ,60G10 ,Mathematics - Probability ,Mathematics - Abstract
As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48:1-141,1987) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary., Comment: Published version at https://doi.org/10.1214/18-AOP1332 in the Annals of Probability by the Institute of Mathematical Statistics
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- 2019
32. Einstein relation and linear response in one-dimensional Mott variable-range hopping
- Author
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Michele Salvi, Alessandra Faggionato, Nina Gantert, Dipartimento di Matematica Guido Castelnuovo, Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], Institut für Mathematische Statistik, Universitaet Muenster, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Università degli Studi di Roma 'La Sapienza' [Rome], Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin] (WIAS), Forschungsverbund Berlin e.V. (FVB) (FVB), Lehrstuhl fur Warscheinlichkeitstheorie, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
- Subjects
Statistics and Probability ,Steady states ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,60K37, 60J25, 60G55, 82D30 ,FOS: Physical sciences ,Mott variable-range hopping ,random walk in random environment ,random conductance model ,environment seen from the particle ,steady states ,linear response ,Einstein relation ,01 natural sciences ,Variable-range hopping ,Environment seen from the particle ,010104 statistics & probability ,60J25 ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Random conductance model ,FOS: Mathematics ,[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] ,0101 mathematics ,Linear response ,Condensed Matter - Statistical Mechanics ,Mathematical Physics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Random walk in random environment ,Statistical Mechanics (cond-mat.stat-mech) ,82D30 ,010102 general mathematics ,Probability (math.PR) ,Mathematical Physics (math-ph) ,ddc ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,60K37 ,60G55 ,Statistics, Probability and Uncertainty ,Humanities ,Mathematics - Probability - Abstract
We consider one-dimensional Mott variable-range hopping with a bias, and prove the linear response as well as the Einstein relation, under an assumption on the exponential moments of the distances between neighboring points. In a previous paper \cite{FGS} we gave conditions on ballisticity, and proved that in the ballistic case the environment viewed from the particle approaches, for almost any initial environment, a given steady state which is absolutely continuous with respect to the original law of the environment. Here, we show that this bias--dependent steady state has a derivative at zero in terms of the bias (linear response), and use this result to get the Einstein relation. Our approach is new: instead of using e.g. perturbation theory or regeneration times, we show that the Radon-Nikodym derivative of the bias--dependent steady state with respect to the equilibrium state in the unbiased case satisfies an $L^p$-bound, $p>2$, uniformly for small bias. This $L^p$-bound yields, by a general argument not involving our specific model, the statement about the linear response., 35 pages, 1 figure
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- 2019
33. Metrics on sets of interval partitions with diversity
- Author
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Douglas Rizzolo, Noah Forman, Soumik Pal, and Matthias Winkel
- Subjects
Statistics and Probability ,interval partition ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,60J25 ,FOS: Mathematics ,Feature (machine learning) ,0101 mathematics ,Topology (chemistry) ,60J60 ,Mathematics ,60J80 ,Stochastic process ,Probability (math.PR) ,Poisson–Dirichlet distribution ,010102 general mathematics ,Metric space ,Alpha (programming language) ,Hausdorff distance ,60G18 ,Metric (mathematics) ,$\alpha $-diversity ,Interval (graph theory) ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,60G52 - Abstract
We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals. Further restricting to interval partitions with alpha-diversity, we then adjust the metric to incorporate diversities. We show that this second metric space is Lusin. An important feature of this topology is that path-continuity in this topology implies the continuous evolution of diversities. This is important in related work on tree-valued stochastic processes where diversities are branch lengths., 12 pages; this preprint generalises and supersedes the parts of arXiv:1609.06706 concerning topologies on interval partitions
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- 2019
34. Continuous-state branching processes, extremal processes and super-individuals
- Author
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Chunhua Ma and Clément Foucart
- Subjects
Statistics and Probability ,60J80 ,Grey martingale ,60G70 ,Super-exponential growth ,010102 general mathematics ,Extremal process ,01 natural sciences ,Infinite variation ,010104 statistics & probability ,Non-linear renormalisation ,Continuous-state branching process ,60G55 ,0101 mathematics ,Statistics, Probability and Uncertainty ,60J80 (Primary), 60G70, 60G55 (Secondary) ,Infinite mean ,Humanities ,Subordinator ,Mathematics - Probability ,Mathematics - Abstract
Les comportements en temps long des flots de processus de branchement en temps et espace continus sont caracterises par des subordinateurs et des processus extremaux. Les processus extremaux apparaissent dans le cas des processus sur-critiques de moyenne infinie et des processus sous-critiques a variation infinie. Les sauts de ces processus extremaux sont interpretes comme des individus initiaux specifiques dont les descendances envahissent la population. Ces individus, qui correspondent aux instants de records d’un certain processus ponctuel de Poisson, sont appeles super-individus. Ils augmentent de facon radicale la vitesse de divergence dans le cas sur-critique et diminuent celle d’extinction dans le cas sous-critique.
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- 2019
35. Infinitely ramified point measures and branching Lévy processes
- Author
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Bastien Mallein, Jean Bertoin, University of Zurich, Institute of Mathematics University of Zurich, Laboratoire Analyse, Géométrie et Applications (LAGA), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)
- Subjects
Statistics and Probability ,Pure mathematics ,Dense set ,Branching Levy process ,Population ,60J80, 60G51, 60G55 ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,Lévy process ,010104 statistics & probability ,510 Mathematics ,Mathematics::Probability ,Branching random walk ,characterization ,1804 Statistics, Probability and Uncertainty ,0101 mathematics ,2613 Statistics and Probability ,growth-fragmentation ,education ,Mathematics ,education.field_of_study ,60J80 ,010102 general mathematics ,Statistics ,Probability and statistics ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,10123 Institute of Mathematics ,Distribution (mathematics) ,Probability and Uncertainty ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,60G51 ,infinitely ramified point measure - Abstract
We call a random point measure infinitely ramified if for every $n\in \mathbb N$, it has the same distribution as the $n$-th generation of some branching random walk. On the other hand, branching L\'evy processes model the evolution of a population in continuous time, such that individuals move in space independently, according to some L\'evy process, and further beget progenies according to some Poissonian dynamics, possibly on an everywhere dense set of times. Our main result connects these two classes of processes much in the same way as in the case of infinitely divisible distributions and L\'evy processes: the value at time $1$ of a branching L\'evy process is an infinitely ramified point measure, and conversely, any infinitely ramified point measure can be obtained as the value at time $1$ of some branching L\'evy process., Comment: To appear in Annals of Probability
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- 2019
36. Asymptotics for Hawkes processes with large and small baseline intensities
- Author
-
Youngsoo Seol
- Subjects
General Mathematics ,law of large numbers ,moderate deviations ,Moment-generating function ,Point process ,central limit theorems ,large deviations ,Law of large numbers ,60F05 ,intensity process ,Large deviations theory ,Statistical physics ,60G55 ,60F10} ,Representation (mathematics) ,Baseline (configuration management) ,Rate function ,Hawkes processes ,Mathematics ,Central limit theorem - Abstract
This paper focuses on asymptotic results for linear Hawkes processes with large and small baseline intensities. The intensity process is one of the main tools used to work with the dynamical properties of a general point process. It is of essential interest in credit risk study, in particular. First, we establish a large deviation principle and a moderate deviation principle for the Hawkes process with large baseline intensity. In addition, a law of large numbers and a central limit theorem are also obtained. Second, we observe asymptotic behaviors for the Hawkes process with small baseline intensity. The main idea of the proof relies on the immigration-birth representation and the observations of the moment generating function for the linear Hawkes process.
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- 2019
37. Convergence of point processes associated with coupon collector's and Dixie cup problems
- Author
-
Andrii Ilienko
- Subjects
Statistics and Probability ,010102 general mathematics ,Probability (math.PR) ,Type (model theory) ,60G55, 60F17 ,01 natural sciences ,Classical limit ,Point process ,010104 statistics & probability ,coupon collector’s problem ,Poisson convergence ,60F17 ,Poisson point process ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,60G55 ,0101 mathematics ,Statistics, Probability and Uncertainty ,Coupon collector's problem ,Dixie cup problem ,poissonization ,Mathematics - Probability ,point processes ,Mathematics - Abstract
We prove that, in the coupon collector's problem, the point processes given by the times of $r$-th arrivals for coupons of each type, centered and normalized in a proper way, converge toward a non-homogeneous Poisson point process. This result is then used to derive some generalizations and infinite-dimensional extensions of classical limit theorems on the topic., Comment: 11 pages
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- 2019
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38. Functional inequalities for marked point processes
- Author
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Ian Flint, Giovanni Luca Torrisi, Nicolas Privault, and School of Physical and Mathematical Sciences
- Subjects
Statistics and Probability ,Mathematics [Science] ,Pure mathematics ,Malliavin calculus ,Poincaré inequality ,Poisson distribution ,Mathematical proof ,01 natural sciences ,Clark-Ocone Formula ,Point process ,marked point processes ,010104 statistics & probability ,symbols.namesake ,60H07 ,Clark-Ocone formula ,FOS: Mathematics ,0101 mathematics ,Representation (mathematics) ,Mathematics ,Laplace transform ,010102 general mathematics ,Probability (math.PR) ,Range (mathematics) ,Poincare inequality ,transportation cost inequalities ,symbols ,Malliavin Calculus ,60G55 ,Statistics, Probability and Uncertainty ,variational representation ,Mathematics - Probability - Abstract
In recent years, a number of functional inequalities have been derived for Poisson random measures, with a wide range of applications. In this paper, we prove that such inequalities can be extended to the setting of marked temporal point processes, under mild assumptions on their Papangelou conditional intensity. First, we derive a Poincaré inequality. Second, we prove two transportation cost inequalities. The first one refers to functionals of marked point processes with a Papangelou conditional intensity and is new even in the setting of Poisson random measures. The second one refers to the law of marked temporal point processes with a Papangelou conditional intensity, and extends a related inequality which is known to hold on a general Poisson space. Finally, we provide a variational representation of the Laplace transform of functionals of marked point processes with a Papangelou conditional intensity. The proofs make use of an extension of the Clark-Ocone formula to marked temporal point processes. Our results are shown to apply to classes of renewal, nonlinear Hawkes and Cox point processes. MOE (Min. of Education, S’pore) Published version
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- 2019
39. Disagreement percolation for the hard-sphere model
- Author
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Hofer Temmel Christoph and Mathematics
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Statistics and Probability ,Phase transition ,absence of phase transition ,82B21 ,82B43 ,FOS: Physical sciences ,Unique gibbs measure ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Stochastic domination ,Lattice (order) ,unique Gibbs measure ,Poisson point process ,FOS: Mathematics ,Absence of phase transition ,Hard-sphere model ,Uniqueness ,Statistical physics ,SDG 7 - Affordable and Clean Energy ,0101 mathematics ,Gibbs measure ,Exponential decay ,stochastic domination ,60D05 ,Condensed Matter - Statistical Mechanics ,Mathematics ,dependent thinning ,Statistical Mechanics (cond-mat.stat-mech) ,Boolean model ,Probability (math.PR) ,010102 general mathematics ,disagreement percolation ,82B21 (60E15 60K35 60G55 82B43 60D05) ,60K35 ,symbols ,60G55 ,Statistics, Probability and Uncertainty ,Disagreement percolation ,60E15 ,Dependent thinning ,hard-sphere model ,Mathematics - Probability ,Cluster expansion - Abstract
Disagreement percolation connects a Gibbs lattice gas and i.i.d. site percolation on the same lattice such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds improve upon known bounds obtained by cluster expansion techniques. The proof uses a novel dependent thinning from a Poisson point process to the hard-sphere model, with the thinning probability related to a derivative of the free energy.
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- 2019
40. Stable limit theorems on the Poisson space
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Ronan Herry, Department of Mathematics - University of Bonn, Rheinische Friedrich-Wilhelms-Universität Bonn, European Project: 694405, RicciBounds, Herry, Ronan, and ERC Advanced Grant Metric measure spaces and Ricci curvature - analytic, geometric, and probabilistic challenges - RicciBounds - 694405 - INCOMING
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Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Gaussian ,Poisson distribution ,Malliavin calculus ,Space (mathematics) ,symbols.namesake ,Poisson point process MSC Classification: 60F15 ,60H07 ,Dimension (vector space) ,Mathematics::Probability ,Théorèmes limites ,60H05 ,60F05 ,Poisson point process ,FOS: Mathematics ,Applied mathematics ,Limit (mathematics) ,Mathematics ,Stochastic process ,Probability (math.PR) ,limit theorems ,MSC Classification: 60F15 ,60G55 ,Convergence stable ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Processus de Poisson ponctuels ,60F15, 60G55, 60H05, 60H07 ,symbols ,Malliavin-Stein ,Statistics, Probability and Uncertainty ,60E10 ,Mathematics - Probability ,stable convergence ,poisson point process - Abstract
We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a 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 another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by Peccati, Sol��, Taqqu & Utzet for Gaussian approximations; and by Peccati for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of D��bler & Peccati. We give an application to stochastic processes., 30 pages, comments are welcome
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- 2019
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41. Probability tilting of compensated fragmentations
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Alexander R. Watson and Quan Shi
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Statistics and Probability ,Class (set theory) ,Work (thermodynamics) ,spine decomposition ,60G51, 60J25, 60J80, 60G55 ,Compensated fragmentation ,Lévy process ,Mathematics::Probability ,60J25 ,FOS: Mathematics ,Decomposition (computer science) ,Statistical physics ,growth-fragmentation ,many-to-one theorem ,Mathematics ,60J80 ,Probability (math.PR) ,compensated fragmentation ,Fragmentation (computing) ,derivative martingale ,Homogeneous ,60G55 ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Mathematics - Probability ,60G51 ,additive martingale - Abstract
Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. These models are widely used in biology, materials science and nuclear physics, and their asymptotic behaviour at large times is interesting both mathematically and practically. The spine decomposition is a key tool in its study. In this work, we consider the class of compensated fragmentations, or homogeneous growth-fragmentations, recently defined by Bertoin. We give a complete spine decomposition of these processes in terms of a L\'evy process with immigration, and apply our result to study the asymptotic properties of the derivative martingale., Comment: 41 pages, 1 figure. This revised version improves the conditions in Theorem 6.1
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- 2019
42. Convergence to scale-invariant Poisson processes and applications in Dickman approximation
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Ilya Molchanov and Chinmoy Bhattacharjee
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Statistics and Probability ,Poisson processes ,scale invariance ,random measures ,Poisson distribution ,Measure (mathematics) ,Point process ,symbols.namesake ,510 Mathematics ,60F05 ,FOS: Mathematics ,11K99 ,Mathematics ,Discrete mathematics ,Sequence ,Weak convergence ,Probability (math.PR) ,Vague convergence ,Probabilistic number theory ,Convergence of random variables ,Dickman distributions ,symbols ,60G57 ,60G55 ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Probability ,360 Social problems & social services - Abstract
We study weak convergence of a sequence of point processes to a scale-invariant simple point process. For a deterministic sequence $(z_n)_{n\in\mathbb{N}}$ of positive real numbers increasing to infinity as $n \to \infty$ and a sequence $(X_k)_{k\in\mathbb{N}}$ of independent non-negative integer-valued random variables, we consider the sequence of point processes \begin{equation*} \nu_n=\sum_{k=1}^\infty X_k \delta_{z_k/z_n}, \quad n\in \mathbb{N}, \end{equation*} and prove that, under some general conditions, it converges vaguely in distribution to a scale-invariant Poisson process $\eta_c$ on $(0,\infty)$ with the intensity measure having the density $ct^{-1}$, $t\in(0,\infty)$. An important motivating example from probabilistic number theory relies on choosing $X_k \sim {\rm Geom}(1-1/p_k)$ and $z_k=\log p_k$, $k\in \mathbb{N}$, where $(p_k)_{k \in \mathbb{N}}$ is an enumeration of the primes in increasing order. We derive a general result on convergence of the integrals $\int_0^1 t \nu_n(dt)$ to the integral $\int_0^1 t \eta_c(dt)$, the latter having a generalized Dickman distribution, thus providing a new way of proving Dickman convergence results. We extend our results to the multivariate setting and provide sufficient conditions for vague convergence in distribution for a broad class of sequences of point processes obtained by mapping the points from $(0,\infty)$ to $\mathbb{R}^d$ via multiplication by i.i.d. random vectors. In addition, we introduce a new class of multivariate Dickman distributions which naturally extends the univariate setting., Comment: Final version, to appear in Electronic Journal of Probability
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- 2019
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43. Modeling non-stationarities in high-frequency financial time series
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Enrico Scalas, Silvano Cincotti, Linda Ponta, Marco Raberto, and Mailan Trinh
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Statistics and Probability ,Monte Carlo method ,Information Criteria ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,FOS: Economics and business ,symbols.namesake ,Stochastic processes ,QA273 ,Bayesian information criterion ,0103 physical sciences ,Statistics ,Compound Poisson process ,Econometrics ,010306 general physics ,High-frequency finance ,Mathematics ,Finance ,Statistical Finance (q-fin.ST) ,business.industry ,Autoregressive conditional duration ,Model selection ,Quantitative Finance - Statistical Finance ,Condensed Matter Physics ,Information criteria ,symbols ,HG0101 ,60G55 ,Akaike information criterion ,HG0106 ,business - Abstract
We study tick-by-tick financial returns belonging to the FTSE MIB index of the Italian Stock Exchange (Borsa Italiana). We can confirm previously detected non-stationarities. However, scaling properties reported in the previous literature for other high-frequency financial data are only approximately valid. As a consequence of the empirical analyses, we propose a simple method for describing non-stationary returns, based on a non-homogeneous normal compound Poisson process. We test this model against the empirical findings and it turns out that the model can approximately reproduce several stylized facts of high-frequency financial time series. Moreover, using Monte Carlo simulations, we analyze order selection for this model class using three information criteria: Akaike's information criterion (AIC), the Bayesian information criterion (BIC) and the Hannan-Quinn information criterion (HQ). For comparison, we also perform a similar Monte Carlo experiment for the ACD (autoregressive conditional duration) model. Our results show that the information criteria work best for small parameter numbers for the compound Poisson type models, whereas for the ACD model the model selection procedure does not work well in certain cases., This version contains an additional analysis on model selection based on information criteria. Version submitted to PLOS ONE
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- 2019
44. On total claim amount for marked Poisson cluster models
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Petra Žugec, Bojan Basrak, Olivier Wintenberger, University of Zagreb, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
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Statistics and Probability ,central limit theorem ,Asymptotic distribution ,Poisson distribution ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Poisson cluster processes ,limit theorems ,Hawkes process ,total claim amount ,stable random variables ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,60F05 ,0502 economics and business ,Statistics ,Cluster (physics) ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Central limit theorem ,050208 finance ,Applied Mathematics ,05 social sciences ,Probability (math.PR) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Distribution (mathematics) ,symbols ,60G55 ,Random variable ,Mathematics - Probability ,stable random variables 2010 MSC: 91B30 - Abstract
We study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence the arrival rate of future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or, alternatively, tends in distribution to an infinite-variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes process as our key example.
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- 2019
45. Multi-scale Lipschitz percolation of increasing events for Poisson random walks
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Alexandre Stauffer, Peter Gracar, Gracar, P., and Stauffer, A.
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Statistics and Probability ,Poisson distribution ,01 natural sciences ,math.PR ,Combinatorics ,010104 statistics & probability ,symbols.namesake ,spread of infection ,Poisson point process ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Space time ,Probability (math.PR) ,010102 general mathematics ,Lipschitz surface ,Lipschitz continuity ,Random walk ,Surface (topology) ,60K35 ,symbols ,Interval (graph theory) ,Multi-scale percolation ,60G55 ,Statistics, Probability and Uncertainty ,Cube ,Mathematics - Probability - Abstract
Consider the graph induced by $\mathbb{Z}^d$, equipped with uniformly elliptic random conductances. At time $0$, place a Poisson point process of particles on $\mathbb{Z}^d$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^d$ and tessellate time into intervals indexed by $\tau$. Given a local event $E(i,\tau)$ that depends only on the particles inside the space time region given by the cube $i$ and the time interval $\tau$, we prove the existence of a Lipschitz connected surface of cells $(i,\tau)$ that separates the origin from infinity on which $E(i,\tau)$ holds. This gives a directly applicable and robust framework for proving results in this setting that need a multi-scale argument. For example, this allows us to prove that an infection spreads with positive speed among the particles., Comment: Additional explanations and figures. arXiv admin note: text overlap with arXiv:1108.6322
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- 2019
46. Universality of local statistics for noncolliding random walks
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Leonid Petrov and Vadim Gorin
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Statistics and Probability ,Pure mathematics ,homogenization ,Markov process ,FOS: Physical sciences ,01 natural sciences ,010104 statistics & probability ,Bernoulli's principle ,symbols.namesake ,FOS: Mathematics ,Ergodic theory ,60C05 ,Mathematics - Combinatorics ,0101 mathematics ,Gibbs measure ,Representation Theory (math.RT) ,Brownian motion ,Mathematical Physics ,Mathematics ,60B20 ,bulk universality ,010102 general mathematics ,Probability (math.PR) ,steepest descent method ,determinantal point processes ,Dyson’s conjecture ,Mathematical Physics (math-ph) ,Noncolliding random walks ,Random walk ,discrete sine process ,Discrete time and continuous time ,Bounded function ,symbols ,60G55 ,Combinatorics (math.CO) ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Mathematics - Representation Theory - Abstract
We consider the $N$-particle noncolliding Bernoulli random walk --- a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some location $x$ (i.e., $x$ is in the "bulk"), and the initial configuration is balanced in a certain sense, then the space-time local statistics at $x$ are asymptotically governed by the extended discrete sine process (which can be identified with a translation invariant ergodic Gibbs measure on lozenge tilings of the plane). We also establish similar results for certain types of random initial data. Our proofs are based on a detailed analysis of the determinantal correlation kernel for the noncolliding Bernoulli random walk. The noncolliding Bernoulli random walk is a discrete analogue of the $\beta=2$ Dyson Brownian Motion whose local statistics are universality governed by the continuous sine process. Our results parallel the ones in the continuous case. In addition, we naturally include situations with inhomogeneous local particle density on scale $T$, which nontrivially affects parameters of the limiting extended sine process, and in a particular case leads to a new behavior., Comment: 59 pages, 12 figures; v2: improved technical details of proofs in section 6
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- 2019
47. Rigidity for zero sets of Gaussian entire functions
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Avner Kiro and Alon Nishry
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Statistics and Probability ,Pure mathematics ,Gaussian ,Entire function ,30B20 ,Gaussian entire functions ,30Dxx, 60G55, 30B20 ,01 natural sciences ,30Dxx ,010104 statistics & probability ,symbols.namesake ,Taylor series ,FOS: Mathematics ,0101 mathematics ,Invariant (mathematics) ,Complex Variables (math.CV) ,point processes ,Complement (set theory) ,Mathematics ,rigidity of linear statistics ,Zero set ,Mathematics - Complex Variables ,010102 general mathematics ,Probability (math.PR) ,Zero (complex analysis) ,Function (mathematics) ,symbols ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane. We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kernel, the zero set is "fully rigid". This means that if we know the location of the zeros in the complement of any given compact set, then the number and location of the zeros inside that set can be determined uniquely. As far as we are aware, this is the first explicit construction in a natural class of random point processes with full rigidity.s with full rigidity., Comment: 12 pages
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- 2019
48. The Widom-Rowlinson model on the Delaunay graph
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Michael Eyers and Stefan Adams
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Statistics and Probability ,Phase transition ,82B21 ,82B43 ,Delaunay tessellation ,random cluster measures ,Computer Science::Computational Geometry ,01 natural sciences ,010104 statistics & probability ,FOS: Mathematics ,Continuum (set theory) ,Statistical physics ,0101 mathematics ,QA ,Representation (mathematics) ,Gibbs measures ,Mathematics ,Connected component ,Particle system ,60G55, 60G57, 82B21, 82B05, 82B26, 82B43 ,Conjecture ,Widom-Rowlinson ,Delaunay triangulation ,Probability (math.PR) ,010102 general mathematics ,mixed site-bond percolation ,phase transition ,multi-body interaction ,coarse graining ,60G57 ,82B26 ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,82B05 ,Potts model - Abstract
We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the length of the Delaunay edges. We show that a phase transition occurs for sufficiently large activities and for sufficiently large potential parameter proving an old conjecture of Lebowitz and Lieb extended to the Delaunay structure. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transition manifests itself in the mixed site-bond percolation of the corresponding random-cluster model. Our proofs rely mainly on geometric properties of Delaunay tessellations in $\mathbb{R}^2 $ and on recent studies [DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is a uniform bound on the number of connected components in the Delaunay graph which provides a novel approach to Delaunay Widom Rowlinson models based on purely geometric arguments. The interaction potential ensures that shorter Delaunay edges are more likely to be open and thus offsets the possibility of having an unbounded number of connected components., 36 pages, 11 figures
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- 2019
49. Pfaffian Schur processes and last passage percolation in a half-quadrant
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Jinho Baik, Guillaume Barraquand, Toufic Suidan, Ivan Corwin, Department of Mathematics - University of Michigan, University of Michigan [Ann Arbor], University of Michigan System-University of Michigan System, and Columbia University [New York]
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Statistics and Probability ,Phase transition ,60K35, 60G55, 82C23, 05E05, 60B20 ,Gaussian ,Diagonal ,Crossover ,FOS: Physical sciences ,Pfaffian ,01 natural sciences ,Point process ,Last passage percolation ,symbols.namesake ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,0103 physical sciences ,KPZ universality class ,05E05 ,FOS: Mathematics ,Statistical physics ,0101 mathematics ,Tracy–Widom distributions ,Condensed Matter - Statistical Mechanics ,Mathematical Physics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,60B20 ,Statistical Mechanics (cond-mat.stat-mech) ,010102 general mathematics ,Probability (math.PR) ,Fredholm Pfaffian ,Schur process ,Mathematical Physics (math-ph) ,Exponential function ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Scaling limit ,phase transition ,60K35 ,symbols ,010307 mathematical physics ,82C23 ,60G55 ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy-Widom distributed, GOE Tracy-Widom distributed, or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy-Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit., Comment: v4: minor typos corrected. v3: 58 pages, to appear in Annals of Probability. The initial submission arXiv:1606.00525v1 has been splitted in two parts. This paper contains only the results on last passage percolation. The results on the facilitated TASEP appear in arXiv:1707.01923
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- 2018
50. Location of the path supremum for self-similar processes with stationary increments
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Yi Shen
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Statistics and Probability ,Path (topology) ,Stationary increment processes ,Probability (math.PR) ,Self-similar processes ,Random locations ,16. Peace & justice ,01 natural sciences ,Infimum and supremum ,60G18 (Primary), 60G55, 60G10 (Secondary) ,010104 statistics & probability ,60G18 ,0103 physical sciences ,FOS: Mathematics ,60G55 ,0101 mathematics ,Statistics, Probability and Uncertainty ,010306 general physics ,60G10 ,Humanities ,Mathematics - Probability ,Mathematics - Abstract
In this paper we consider the distribution of the location of the path supremum in a fixed interval for self-similar processes with stationary increments. To this end, a point process is constructed and its relation to the distribution of the location of the path supremum is studied. Using this framework, we show that the distribution has a spectral-type representation, in the sense that it is always a mixture of a special group of absolutely continuous distributions, plus point masses on the two boundaries. Bounds on the value and the derivatives of the density function are established. We further discuss self-similar L\'{e}vy processes as an example. Most of the results in this paper can be generalized to a group of random locations, including the location of the largest jump, etc., Comment: 13 pages
- Published
- 2018
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