We show that the number of copies of a given rooted tree in a conditioned Galton-Watson tree satisfies a law of large numbers under a minimal moment condition on the offspring distribution.
我们证明了条件高尔顿-沃森树中给定根树的拷贝数在子代分布的最小矩条件下满足大数定律。
{"title":"On general subtrees of a conditioned Galton–Watson tree","authors":"S. Janson","doi":"10.1214/21-ECP392","DOIUrl":"https://doi.org/10.1214/21-ECP392","url":null,"abstract":"We show that the number of copies of a given rooted tree in a conditioned Galton-Watson tree satisfies a law of large numbers under a minimal moment condition on the offspring distribution.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75996964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-08DOI: 10.1016/j.jfa.2021.109225
A. Ohashi, F. Russo, E. Shamarova
{"title":"Smoothness of densities for path-dependent SDEs under Hörmander's condition","authors":"A. Ohashi, F. Russo, E. Shamarova","doi":"10.1016/j.jfa.2021.109225","DOIUrl":"https://doi.org/10.1016/j.jfa.2021.109225","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73311010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a two-type reducible branching Brownian motion, defined as a two type branching particle system on the real line, in which particles of type $1$ can give birth to particles of type $2$, but not reciprocally. This process has been shown by Biggins to exhibit an anomalous spreading behaviour under specific conditions: in that situation, the rightmost particle at type $t$ is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type $1$ or of type $2$. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer. The aim of this article is to refine the previous results and study the asymptotic behaviour of the extremal process of the two-type reducible branching Brownian motion. If the branching Brownian motion exhibits an anomalous spreading behaviour, its asymptotic differs from what it typically expected in branching Brownian motions.
{"title":"Anomalous spreading in reducible multitype branching Brownian motion","authors":"M. Belloum, Bastien Mallein","doi":"10.1214/21-EJP629","DOIUrl":"https://doi.org/10.1214/21-EJP629","url":null,"abstract":"We consider a two-type reducible branching Brownian motion, defined as a two type branching particle system on the real line, in which particles of type $1$ can give birth to particles of type $2$, but not reciprocally. This process has been shown by Biggins to exhibit an anomalous spreading behaviour under specific conditions: in that situation, the rightmost particle at type $t$ is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type $1$ or of type $2$. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer. The aim of this article is to refine the previous results and study the asymptotic behaviour of the extremal process of the two-type reducible branching Brownian motion. If the branching Brownian motion exhibits an anomalous spreading behaviour, its asymptotic differs from what it typically expected in branching Brownian motions.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89274371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marcelo Campos, Matthew Jenssen, Marcus Michelen, J. Sahasrabudhe
Let $M_n$ be drawn uniformly from all $pm 1$ symmetric $n times n$ matrices. We show that the probability that $M_n$ is singular is at most $exp(-c(nlog n)^{1/2})$, which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of $exp(-c n^{1/2})$ on the singularity probability, our method is different and considerably simpler.
设$M_n$从所有的$pm 1$对称的$n times n$矩阵中均匀地画出来。我们证明$M_n$是奇异的概率最多是$exp(-c(nlog n)^{1/2})$，这代表了在最近解决这个问题的方法中的一个自然障碍。除了改进了最著名的Campos, Mattos, Morris和Morrison ($exp(-c n^{1/2})$)关于奇点概率的上界之外，我们的方法是不同的，而且简单得多。
{"title":"Singularity of random symmetric matrices revisited","authors":"Marcelo Campos, Matthew Jenssen, Marcus Michelen, J. Sahasrabudhe","doi":"10.1090/proc/15807","DOIUrl":"https://doi.org/10.1090/proc/15807","url":null,"abstract":"Let $M_n$ be drawn uniformly from all $pm 1$ symmetric $n times n$ matrices. We show that the probability that $M_n$ is singular is at most $exp(-c(nlog n)^{1/2})$, which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of $exp(-c n^{1/2})$ on the singularity probability, our method is different and considerably simpler.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73580925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hermon and Hutchcroft have recently proved the long-standing conjecture that in Bernoulli(p) bond percolation on any nonamenable transitive graph G, at any p > p_c(G), the probability that the cluster of the origin is finite but has a large volume n decays exponentially in n. A corollary is that all infinite clusters have anchored expansion almost surely. They have asked if these results could hold more generally, for any finite energy ergodic invariant percolation. We give a counterexample, an invariant percolation on the 4-regular tree.
{"title":"Finite-energy infinite clusters without anchored expansion.","authors":"G. Pete, 'Ad'am Tim'ar","doi":"10.3150/20-BEJ1311","DOIUrl":"https://doi.org/10.3150/20-BEJ1311","url":null,"abstract":"Hermon and Hutchcroft have recently proved the long-standing conjecture that in Bernoulli(p) bond percolation on any nonamenable transitive graph G, at any p > p_c(G), the probability that the cluster of the origin is finite but has a large volume n decays exponentially in n. A corollary is that all infinite clusters have anchored expansion almost surely. They have asked if these results could hold more generally, for any finite energy ergodic invariant percolation. We give a counterexample, an invariant percolation on the 4-regular tree.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82955243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Herein, a methodology is developed to replicate functions, measures and stochastic processes onto a compact metric space. Many results are easily established for the replica objects and then transferred back to the original ones. Two problems are solved within to demonstrate the method: (1) Finite-dimensional convergence for processes living on general topological spaces. (2) New tightness and relative compactness criteria are given for the Skorokhod space $D(mathbf{R}^{+};E)$ with $E$ being a general Tychonoff space. The methods herein are also used in companion papers to establish the: (3) existence of, uniqueness of and convergence to martingale problem solutions, (4) classical Fujisaki-Kallianpur-Kunita and Duncan-Mortensen-Zakai filtering equations and stationary filters, (5) finite-dimensional convergence to stationary signal-filter pairs, (6) invariant measures of Markov processes, and (7) Ray-Knight theory all in general settings.
{"title":"Replication and Its Application to Weak Convergence","authors":"C. Dong, M. Kouritzin","doi":"10.7939/R3K06XF1W","DOIUrl":"https://doi.org/10.7939/R3K06XF1W","url":null,"abstract":"Herein, a methodology is developed to replicate functions, measures and stochastic processes onto a compact metric space. Many results are easily established for the replica objects and then transferred back to the original ones. Two problems are solved within to demonstrate the method: (1) Finite-dimensional convergence for processes living on general topological spaces. (2) New tightness and relative compactness criteria are given for the Skorokhod space $D(mathbf{R}^{+};E)$ with $E$ being a general Tychonoff space. The methods herein are also used in companion papers to establish the: (3) existence of, uniqueness of and convergence to martingale problem solutions, (4) classical Fujisaki-Kallianpur-Kunita and Duncan-Mortensen-Zakai filtering equations and stationary filters, (5) finite-dimensional convergence to stationary signal-filter pairs, (6) invariant measures of Markov processes, and (7) Ray-Knight theory all in general settings.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77510203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the barrier function in Root's solution to the Skorokhod embedding problem is continuous and finite at every point where the target measure has no atom and its absolutely continuous part is locally bounded away from zero.
{"title":"On the continuity of the root barrier","authors":"Erhan Bayraktar, Thomas Bernhardt","doi":"10.1090/proc/15765","DOIUrl":"https://doi.org/10.1090/proc/15765","url":null,"abstract":"We show that the barrier function in Root's solution to the Skorokhod embedding problem is continuous and finite at every point where the target measure has no atom and its absolutely continuous part is locally bounded away from zero.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87612588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that random triangulations of types I, II, and III with a simple boundary under the critical Boltzmann weight converge in the scaling limit to the Brownian disk. The proof uses a bijection due to Poulalhon and Schaeffer between type III triangulations of the $p$-gon and so-called blossoming forests. A variant of this bijection was also used by Addario-Berry and the first author to prove convergence of type III triangulations to the Brownian map, but new ideas are needed to handle the simple boundary. Our result is an ingredient in the program of the second and third authors on the convergence of uniform triangulations under the Cardy embedding.
{"title":"Scaling limit of triangulations of polygons","authors":"M. Albenque, N. Holden, Xin Sun","doi":"10.1214/20-ejp537","DOIUrl":"https://doi.org/10.1214/20-ejp537","url":null,"abstract":"We prove that random triangulations of types I, II, and III with a simple boundary under the critical Boltzmann weight converge in the scaling limit to the Brownian disk. The proof uses a bijection due to Poulalhon and Schaeffer between type III triangulations of the $p$-gon and so-called blossoming forests. A variant of this bijection was also used by Addario-Berry and the first author to prove convergence of type III triangulations to the Brownian map, but new ideas are needed to handle the simple boundary. Our result is an ingredient in the program of the second and third authors on the convergence of uniform triangulations under the Cardy embedding.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79126674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a Jackson network with regenerative input flows in which every server is subject to a random environment influence generating breakdowns and repairs. They occur in accordance with two independent sequences of i.i.d. random variables. We establish a theorem on the strong approximation of the vector of queue lengths by a reflected Brownian motion in positive orthant.
{"title":"Jackson network in a random environment: strong approximation","authors":"E. Bashtova, E. Lenena","doi":"10.47910/femj202015","DOIUrl":"https://doi.org/10.47910/femj202015","url":null,"abstract":"We consider a Jackson network with regenerative input flows in which every server is subject to a random environment influence generating breakdowns and repairs. They occur in accordance with two independent sequences of i.i.d. random variables. We establish a theorem on the strong approximation of the vector of queue lengths by a reflected Brownian motion in positive orthant.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86851497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-27DOI: 10.2422/2036-2145.202011_016
M. Fathi, Dan Mikulincer
We investigate stability of invariant measures of diffusion processes with respect to $L^p$ distances on the coefficients, under an assumption of log-concavity. The method is a variant of a technique introduced by Crippa and De Lellis to study transport equations. As an application, we prove a partial extension of an inequality of Ledoux, Nourdin and Peccati relating transport distances and Stein discrepancies to a non-Gaussian setting via the moment map construction of Stein kernels.
{"title":"Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels","authors":"M. Fathi, Dan Mikulincer","doi":"10.2422/2036-2145.202011_016","DOIUrl":"https://doi.org/10.2422/2036-2145.202011_016","url":null,"abstract":"We investigate stability of invariant measures of diffusion processes with respect to $L^p$ distances on the coefficients, under an assumption of log-concavity. The method is a variant of a technique introduced by Crippa and De Lellis to study transport equations. As an application, we prove a partial extension of an inequality of Ledoux, Nourdin and Peccati relating transport distances and Stein discrepancies to a non-Gaussian setting via the moment map construction of Stein kernels.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79580043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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