{"title":"Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes","authors":"G. Torrisi","doi":"10.1214/15-AIHP730","DOIUrl":"https://doi.org/10.1214/15-AIHP730","url":null,"abstract":"","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"62 1","pages":"679-700"},"PeriodicalIF":1.5,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79199756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we obtain scaling limits of Λ-coalescents near time zero under a regularly varying assumption. In particular this covers the case of Kingman’s coalescent and beta coalescents. The limiting processes are coalescents with infinite mass, obtained geometrically as tangent cones of Evans metric space associated with the coalescent. In the case of Kingman’s coalescent we are able to obtain a simple construction of the limiting space using a two-sided Brownian motion.
{"title":"Scaling limits of coalescent processes near time zero","authors":"Batı Şengül","doi":"10.1214/15-AIHP727","DOIUrl":"https://doi.org/10.1214/15-AIHP727","url":null,"abstract":"In this paper we obtain scaling limits of Λ-coalescents near time zero under a regularly varying assumption. In particular this covers the case of Kingman’s coalescent and beta coalescents. The limiting processes are coalescents with infinite mass, obtained geometrically as tangent cones of Evans metric space associated with the coalescent. In the case of Kingman’s coalescent we are able to obtain a simple construction of the limiting space using a two-sided Brownian motion.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"51 1","pages":"616-640"},"PeriodicalIF":1.5,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87340910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric. We discuss the advantages of this approach, over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart, as the sample size tends to infinity. A key property in the study of GPCA is the isometry between the Wasserstein space and a closed convex subset of the space of square-integrable functions, with respect to an appropriate measure. Therefore, we consider the general problem of PCA in a closed convex subset of a separable Hilbert space, which serves as basis for the analysis of GPCA and also has interest in its own right. We provide illustrative examples on simple statistical models, to show the benefits of this approach for data analysis. The method is also applied to a real dataset of population pyramids.
{"title":"Geodesic PCA in the Wasserstein space by Convex PCA","authors":"Jérémie Bigot, R. Gouet, T. Klein, Alfredo López","doi":"10.1214/15-AIHP706","DOIUrl":"https://doi.org/10.1214/15-AIHP706","url":null,"abstract":"We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric. We discuss the advantages of this approach, over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart, as the sample size tends to infinity. A key property in the study of GPCA is the isometry between the Wasserstein space and a closed convex subset of the space of square-integrable functions, with respect to an appropriate measure. Therefore, we consider the general problem of PCA in a closed convex subset of a separable Hilbert space, which serves as basis for the analysis of GPCA and also has interest in its own right. We provide illustrative examples on simple statistical models, to show the benefits of this approach for data analysis. The method is also applied to a real dataset of population pyramids.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"7 1","pages":"1-26"},"PeriodicalIF":1.5,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73146535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Let W λ,b (x) = (cid:2) ∞ n = 0 λ n g(b n x) where b ≥ 2 is an integer and g(u) = cos ( 2 πu) (classical Weierstrass function). Building on work by Ledrappier (In Symbolic Dynamics and Its Applications (1992) 285–293), Bara´nski, Bárány and Romanowska ( Adv. Math. 265 (2014) 32–59) and Tsujii ( Nonlinearity 14 (2001) 1011–1027), we provide an elementary proof that the Hausdorff dimension of W λ,b equals 2 + log λ log b for all λ ∈ (λ b , 1 ) with a suitable λ b < 1. This reproduces results by Bara´nski, Bárány and Romanowska ( Adv. Math. 265 (2014) 32–59) without using the dimension theory for hyperbolic measures of Ledrappier and Young ( Ann. of Math. (2) 122 (1985) 540–574; Comm. Math. Phys. 117 (1988) 529–548), which is replaced by a simple telescoping argument together with a recursive multi-scale estimate. Résumé. (In Symbolic Dynamics and Its Applications (1992) 285–293), de Bara´nski, Bárány et Romanowska ( Adv. Math. 265 (2014) 32–59) et de Tsujii ( Nonlinearity 14 (2001) 1011–1027), nous présentons une démonstra-tion élémentaire du fait que la dimension de Hausdorff de W λ,b est égale à 2 + log λ log b pour tout λ ∈ (λ b , 1 ) avec λ b < 1 approprié. Cela reproduit des résultats de Bara´nski, Bárány et Romanowska ( Adv. Math. 265 (2014) 32–59) sans utiliser la théorie de dimension des mesures hyperboliques de Ledrappier et Young ( Ann. of Math. (2) 122 (1985) 540–574 ; Comm. Math. Phys. 117 (1988) 529–548), laquelle est remplacée par un argument téléscopique élémentaire conjointement avec une estimation récursive multi-échelle.
． 设W λ，b (x) = (cid:2)∞n = 0 λ n g(b n x)，其中b≥2为整数，g(u) = cos (2 πu)(经典Weierstrass函数)。在Ledrappier (In Symbolic Dynamics and Its Applications(1992) 285-293)、Bara´nski, Bárány和Romanowska (Adv. Math. 265(2014) 32-59)和Tsujii (Nonlinearity 14(2001) 1011-1027)的工作基础上，我们提供了一个初等证明，证明对于所有λ∈(λ b, 1)且λ b < 1的情况下，W λ，b的Hausdorff维数等于2 + log λ log b。这再现了Bara´nski, Bárány和Romanowska (Adv. Math. 265(2014) 32-59)的结果，而没有使用Ledrappier和Young (Ann.)的双曲测量的维数理论。的数学。(2) 122 (1985) 540-574;通讯。数学。物理学报，117(1988)529-548)，它被一个简单的伸缩论证和一个递归的多尺度估计所取代。的简历。(In Symbolic Dynamics and Its Applications (1992) 285-293)， de Bara ' nski, Bárány et Romanowska (Adv. Math. 265 (2014) 32-59) et de Tsujii (Nonlinearity 14 (2001) 1011-1027)， nous pracentsons one dsammonstrage du fait que la dimension de Hausdorff de W λ，b est + log λ log b pour tout λ∈(λ b, 1) avec λ b < 1)Cela redududes recametssulats de Bara ' nski, Bárány et Romanowska (Adv. Math. 265 (2014) 32-59)， sans utiliser la thcametyde dimensionesdes measures hyperbolques de Ledrappier et Young (Ann。的数学。(2) 122 (1985) 540-574;通讯。数学。物理学报，117 (1988)529-548)，laquelle est取代了same paran论点，即sami - sami - sami - sami - sami - sami - sami - sami - sami - sami。
{"title":"A simpler proof for the dimension of the graph of the classical Weierstrass function","authors":"G. Keller","doi":"10.1214/15-AIHP711","DOIUrl":"https://doi.org/10.1214/15-AIHP711","url":null,"abstract":". Let W λ,b (x) = (cid:2) ∞ n = 0 λ n g(b n x) where b ≥ 2 is an integer and g(u) = cos ( 2 πu) (classical Weierstrass function). Building on work by Ledrappier (In Symbolic Dynamics and Its Applications (1992) 285–293), Bara´nski, Bárány and Romanowska ( Adv. Math. 265 (2014) 32–59) and Tsujii ( Nonlinearity 14 (2001) 1011–1027), we provide an elementary proof that the Hausdorff dimension of W λ,b equals 2 + log λ log b for all λ ∈ (λ b , 1 ) with a suitable λ b < 1. This reproduces results by Bara´nski, Bárány and Romanowska ( Adv. Math. 265 (2014) 32–59) without using the dimension theory for hyperbolic measures of Ledrappier and Young ( Ann. of Math. (2) 122 (1985) 540–574; Comm. Math. Phys. 117 (1988) 529–548), which is replaced by a simple telescoping argument together with a recursive multi-scale estimate. Résumé. (In Symbolic Dynamics and Its Applications (1992) 285–293), de Bara´nski, Bárány et Romanowska ( Adv. Math. 265 (2014) 32–59) et de Tsujii ( Nonlinearity 14 (2001) 1011–1027), nous présentons une démonstra-tion élémentaire du fait que la dimension de Hausdorff de W λ,b est égale à 2 + log λ log b pour tout λ ∈ (λ b , 1 ) avec λ b < 1 approprié. Cela reproduit des résultats de Bara´nski, Bárány et Romanowska ( Adv. Math. 265 (2014) 32–59) sans utiliser la théorie de dimension des mesures hyperboliques de Ledrappier et Young ( Ann. of Math. (2) 122 (1985) 540–574 ; Comm. Math. Phys. 117 (1988) 529–548), laquelle est remplacée par un argument téléscopique élémentaire conjointement avec une estimation récursive multi-échelle.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"310 1","pages":"169-181"},"PeriodicalIF":1.5,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79693641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, we prove pathwise uniqueness for stochastic degenerate systems with a Hölder drift, for a Hölder exponent larger than the critical value 2 / 3. This work extends to the degenerate setting the earlier results obtained by Zvonkin ( Mat. Sb. (N.S.) 93(135) (1974) 129–149, 152), Veretennikov ( Mat. Sb. (N.S.) 111(153) (1980) 434–452, 480), Krylov and Röckner ( Probab. Theory Related Fields 131 (2) (2005) 154–196) from non-degenerate to degenerate cases. The existence of a threshold for the Hölder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on regularization properties of the associated PDE, which is degenerate in the current framework and is based on a parametrix method. Résumé. ( L’apparition d’un seuil critique pour l’exposant peut-être vue comme le prix à payer pour la dégénérescence. La preuve repose sur des résultats de régularité de la solution de l’EDP associée, qui est dégénérée, et est basée sur une méthode parametrix.
{"title":"Strong existence and uniqueness for degenerate SDE with Hölder drift","authors":"P. C. D. Raynal","doi":"10.1214/15-AIHP716","DOIUrl":"https://doi.org/10.1214/15-AIHP716","url":null,"abstract":". In this paper, we prove pathwise uniqueness for stochastic degenerate systems with a Hölder drift, for a Hölder exponent larger than the critical value 2 / 3. This work extends to the degenerate setting the earlier results obtained by Zvonkin ( Mat. Sb. (N.S.) 93(135) (1974) 129–149, 152), Veretennikov ( Mat. Sb. (N.S.) 111(153) (1980) 434–452, 480), Krylov and Röckner ( Probab. Theory Related Fields 131 (2) (2005) 154–196) from non-degenerate to degenerate cases. The existence of a threshold for the Hölder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on regularization properties of the associated PDE, which is degenerate in the current framework and is based on a parametrix method. Résumé. ( L’apparition d’un seuil critique pour l’exposant peut-être vue comme le prix à payer pour la dégénérescence. La preuve repose sur des résultats de régularité de la solution de l’EDP associée, qui est dégénérée, et est basée sur une méthode parametrix.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"34 1","pages":"259-286"},"PeriodicalIF":1.5,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81504005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cet article analyse une classe de methodes de Monte Carlo avancees de type particulaire introduites par Andrieu, Doucet, et Holenstein (J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (2010) 1–269). Nous presentons une interpretation naturelle de ces methodes en termes de mesures de Feynman–Kac particulaires non biaisees classiques et d’une nouvelle formule de dualite entre modeles de Feynman–Kac. Ce cadre d’etude apporte un nouvel eclairage sur les fondations et l’analyse mathematique de ces methodes. Une consequence importante est l’equivalence de ces dernieres avec la methode d’echantillonnage de Gibbs d’une distribution de Feynman–Kac multi-corps. Notre etude developpe aussi un nouveau calcul differentiel stochastique fonde sur des techniques geometriques et combinatoires. Ces techniques permettent d’obtenir des developpements non asymptotiques des semigroupes de modeles de Monte Carlo par Chaines de Markov particulaires autour de leur mesure invariante, en fonction de la taille des systemes de particules en interaction auxiliaires. Cette analyse conduit a des estimations quantitatives precises de la convergence a l’equilibre de ces modeles par rapport a l’horizon temporel et la taille des systemes. Nous illustrons ces resultats avec quelques implications directes, notamment l’estimation precise des coefficients de contraction et des exposants de Lyapunov de ces algorithmes de simulation, ainsi que l’estimation fine de l’erreur en norme $mathbb{L}_{p}$ entre la loi des etats aleatoires de ces chaines de Markov et leur mesure d’equilibre. Le cadre abstrait de l’article permet d’elaborer et d’etendre de facon naturelle ces methodes a des classes d’algorithmes fondes sur des evolutions d’ilots particulaires (aussi connus sous le nom $mathrm{SMC}^{2}$). Nous montrons enfin comment ce cadre general et les resultats de l’article s’appliquent a l’etude de problemes de filtrage non lineaire, l’estimation de parametres fixes dans des modeles de chaines de Markov cachees, et dans des problemes d’integration trajectorielle rencontres en physique quantique et en chimie moleculaire.
{"title":"On particle Gibbs samplers","authors":"P. Moral, R. Kohn, F. Patras","doi":"10.1214/15-AIHP695","DOIUrl":"https://doi.org/10.1214/15-AIHP695","url":null,"abstract":"Cet article analyse une classe de methodes de Monte Carlo avancees de type particulaire introduites par Andrieu, Doucet, et Holenstein (J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (2010) 1–269). Nous presentons une interpretation naturelle de ces methodes en termes de mesures de Feynman–Kac particulaires non biaisees classiques et d’une nouvelle formule de dualite entre modeles de Feynman–Kac. Ce cadre d’etude apporte un nouvel eclairage sur les fondations et l’analyse mathematique de ces methodes. Une consequence importante est l’equivalence de ces dernieres avec la methode d’echantillonnage de Gibbs d’une distribution de Feynman–Kac multi-corps. Notre etude developpe aussi un nouveau calcul differentiel stochastique fonde sur des techniques geometriques et combinatoires. Ces techniques permettent d’obtenir des developpements non asymptotiques des semigroupes de modeles de Monte Carlo par Chaines de Markov particulaires autour de leur mesure invariante, en fonction de la taille des systemes de particules en interaction auxiliaires. Cette analyse conduit a des estimations quantitatives precises de la convergence a l’equilibre de ces modeles par rapport a l’horizon temporel et la taille des systemes. Nous illustrons ces resultats avec quelques implications directes, notamment l’estimation precise des coefficients de contraction et des exposants de Lyapunov de ces algorithmes de simulation, ainsi que l’estimation fine de l’erreur en norme $mathbb{L}_{p}$ entre la loi des etats aleatoires de ces chaines de Markov et leur mesure d’equilibre. Le cadre abstrait de l’article permet d’elaborer et d’etendre de facon naturelle ces methodes a des classes d’algorithmes fondes sur des evolutions d’ilots particulaires (aussi connus sous le nom $mathrm{SMC}^{2}$). Nous montrons enfin comment ce cadre general et les resultats de l’article s’appliquent a l’etude de problemes de filtrage non lineaire, l’estimation de parametres fixes dans des modeles de chaines de Markov cachees, et dans des problemes d’integration trajectorielle rencontres en physique quantique et en chimie moleculaire.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"27 1","pages":"1687-1733"},"PeriodicalIF":1.5,"publicationDate":"2016-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89838083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If $h_*$ and $u_*$ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that $h_* 0$.
{"title":"Level-set percolation for the Gaussian free field on a transient tree","authors":"Angelo Abacherli, A. Sznitman","doi":"10.1214/16-AIHP799","DOIUrl":"https://doi.org/10.1214/16-AIHP799","url":null,"abstract":"We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If $h_*$ and $u_*$ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that $h_* 0$.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"2 1","pages":"173-201"},"PeriodicalIF":1.5,"publicationDate":"2016-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82031076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide non-asymptotic $L^1$ bounds to the normal for four well-known models in statistical physics and particle systems in $mathbb{Z}^d$; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process. In the Ising model, we obtain an $L^1$ distance bound between the total magnetization and the normal distribution at any temperature when the magnetic moment parameter is nonzero, and when the inverse temperature is below critical and the magnetic moment parameter is zero. In the percolation model we obtain such a bound for the total number of points in a finite region belonging to an infinite cluster in dimensions $d ge 2$, in the voter model for the occupation time of the origin in dimensions $d ge 7$, and for finite time integrals of non-constant increasing cylindrical functions evaluated on the one dimensional supercritical contact process started in its unique invariant distribution. The tool developed for these purposes is a version of Stein's method adapted to positively associated random variables. In one dimension, letting $boldsymbol{xi}=(xi_1,ldots,xi_m)$ be a positively associated mean zero random vector with components that obey the bound $|xi_i| le B, i=1,ldots,m$, and whose sum $W = sum_{i=1}^m xi_i$ has variance 1, it holds that $$ d_1 left(mathcal{L}(W),mathcal{L}(Z) right) leq 5B + sqrt{frac{8}{pi}}sum_{i neq j} mathbb{E}[xi_i xi_j] $$ where $Z$ has the standard normal distribution and $d_1(cdot,cdot)$ is the $L^1$ metric. Our methods apply in the multidimensional case with the $L^1$ metric replaced by a smooth function metric.
{"title":"Stein’s method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process","authors":"L. Goldstein, Nathakhun Wiroonsri","doi":"10.1214/16-AIHP808","DOIUrl":"https://doi.org/10.1214/16-AIHP808","url":null,"abstract":"We provide non-asymptotic $L^1$ bounds to the normal for four well-known models in statistical physics and particle systems in $mathbb{Z}^d$; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process. In the Ising model, we obtain an $L^1$ distance bound between the total magnetization and the normal distribution at any temperature when the magnetic moment parameter is nonzero, and when the inverse temperature is below critical and the magnetic moment parameter is zero. In the percolation model we obtain such a bound for the total number of points in a finite region belonging to an infinite cluster in dimensions $d ge 2$, in the voter model for the occupation time of the origin in dimensions $d ge 7$, and for finite time integrals of non-constant increasing cylindrical functions evaluated on the one dimensional supercritical contact process started in its unique invariant distribution. \u0000The tool developed for these purposes is a version of Stein's method adapted to positively associated random variables. In one dimension, letting $boldsymbol{xi}=(xi_1,ldots,xi_m)$ be a positively associated mean zero random vector with components that obey the bound $|xi_i| le B, i=1,ldots,m$, and whose sum $W = sum_{i=1}^m xi_i$ has variance 1, it holds that $$ d_1 left(mathcal{L}(W),mathcal{L}(Z) right) leq 5B + sqrt{frac{8}{pi}}sum_{i neq j} mathbb{E}[xi_i xi_j] $$ where $Z$ has the standard normal distribution and $d_1(cdot,cdot)$ is the $L^1$ metric. Our methods apply in the multidimensional case with the $L^1$ metric replaced by a smooth function metric.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"5 1","pages":"385-421"},"PeriodicalIF":1.5,"publicationDate":"2016-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90145181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove that certain frog model statistics are monotone in the initial configuration for two nonstandard stochastic dominance relations: the increasing concave and the probability generating function orders. This extends many canonical theorems. We connect recurrence for random initial configurations to recurrence for deterministic configurations. Also, the limiting shape of activated sites on the integer lattice respects both of these orders. Other implications include monotonicity results on transience of the frog model where the number of frogs per vertex decays away from the origin, on survival of the frog model with death, and on the time to visit a given vertex in any frog model.
{"title":"Stochastic orders and the frog model","authors":"Tobias Johnson, M. Junge","doi":"10.1214/17-AIHP830","DOIUrl":"https://doi.org/10.1214/17-AIHP830","url":null,"abstract":"The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove that certain frog model statistics are monotone in the initial configuration for two nonstandard stochastic dominance relations: the increasing concave and the probability generating function orders. \u0000This extends many canonical theorems. We connect recurrence for random initial configurations to recurrence for deterministic configurations. Also, the limiting shape of activated sites on the integer lattice respects both of these orders. Other implications include monotonicity results on transience of the frog model where the number of frogs per vertex decays away from the origin, on survival of the frog model with death, and on the time to visit a given vertex in any frog model.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"48 1","pages":"1013-1030"},"PeriodicalIF":1.5,"publicationDate":"2016-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89140556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We compute the generating series for the simplest class of bi-free cumulants, beyond the free cumulants, the two-bands bi-free cumulants of a pair of a left and a right variable. We also consider two-faced systems with a commutation condition implying that two-bands moments, that is expectation values of the product of a monomial of left and a monomial of right variables determine the other moments. Examples include hyponormal operators, dual systems in free entropy theory and bi-partite systems. Résumé. Pour une paire de variables, une gauche et une droite, nous calculons la série génératrice des plus simples cumulants bi-libres au-delà des cumulants libres. On considère aussi des systèmes à deux faces, satisfaisant une condition de commutation qui assure que les moments à deux bandes, c’est-à-dire les espérances des produits d’un monôme de variables gauches et d’un monôme de variables droites, déterminent tous les autres moments non-commutatifs. Les opérateurs hyponormaux, les systèmes duaux de la théorie de l’entropie libre, ainsi que les systèmes bipartites, fournissent des exemples.
． We在线使用发电series for the纪念class of bi-free cumulants、beyond the free cumulants the two-bands bi-free cumulants可变of a pair of a left and right。We also联同two-faced systems with a开关implying that two-bands时光,只要瞧广州改良of the products of a monomial of a left and monomial of right ?变量确定the other时光。实例包括二元hyponormal运营商,free entropy systems in theory and bi-partite systems。摘要。对于一对变量，一个左，一个右，我们计算生成最简单的双自由累积数超过自由累积数的级数。我们还考虑了满足切换条件的双面系统，该条件保证了双波段力矩，即单个左变量和单个右变量的乘积的期望，决定了所有其他非切换力矩。例子包括下正则算子、自由熵理论的对偶系统和二部系统。
{"title":"Free probability for pairs of faces II: $2$-variables bi-free partial $R$-transform and systems with rank $le1$ commutation","authors":"D. Voiculescu","doi":"10.1214/14-AIHP623","DOIUrl":"https://doi.org/10.1214/14-AIHP623","url":null,"abstract":". We compute the generating series for the simplest class of bi-free cumulants, beyond the free cumulants, the two-bands bi-free cumulants of a pair of a left and a right variable. We also consider two-faced systems with a commutation condition implying that two-bands moments, that is expectation values of the product of a monomial of left and a monomial of right variables determine the other moments. Examples include hyponormal operators, dual systems in free entropy theory and bi-partite systems. Résumé. Pour une paire de variables, une gauche et une droite, nous calculons la série génératrice des plus simples cumulants bi-libres au-delà des cumulants libres. On considère aussi des systèmes à deux faces, satisfaisant une condition de commutation qui assure que les moments à deux bandes, c’est-à-dire les espérances des produits d’un monôme de variables gauches et d’un monôme de variables droites, déterminent tous les autres moments non-commutatifs. Les opérateurs hyponormaux, les systèmes duaux de la théorie de l’entropie libre, ainsi que les systèmes bipartites, fournissent des exemples.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"13 1","pages":"1-15"},"PeriodicalIF":1.5,"publicationDate":"2016-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82050496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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