We provide an elementary proof of the support of the law of a hypoelliptic Brownian motion on the Heisenberg group $mathbb{H}$. We consider a control norm associated to left-invariant vector fields on $mathbb{H}$, and describe the support in terms of the space of finite energy horizontal curves.
{"title":"On the support of a hypoelliptic diffusion on the Heisenberg group","authors":"M. Carfagnini","doi":"10.30757/alea.v20-26","DOIUrl":"https://doi.org/10.30757/alea.v20-26","url":null,"abstract":"We provide an elementary proof of the support of the law of a hypoelliptic Brownian motion on the Heisenberg group $mathbb{H}$. We consider a control norm associated to left-invariant vector fields on $mathbb{H}$, and describe the support in terms of the space of finite energy horizontal curves.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48746002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first formulation in Cooper and Frieze (2005), and then used and refined in a list of papers by Cooper, Frieze and coauthors. We work in the original setting, considering a growing sequence of irreducible Markov chains on n states. We assume that the chain is rapidly mixing and with a stationary measure with no entry being either too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state x—for the chain started at stationarity—up to a small multiplicative correction. While the proof by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob’s transform of the chain on the complement of the state x.
{"title":"A probabilistic proof of Cooper and Frieze's First Visit Time Lemma","authors":"F. Manzo, Matteo Quattropani, E. Scoppola","doi":"10.30757/alea.v18-64","DOIUrl":"https://doi.org/10.30757/alea.v18-64","url":null,"abstract":"We present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first formulation in Cooper and Frieze (2005), and then used and refined in a list of papers by Cooper, Frieze and coauthors. We work in the original setting, considering a growing sequence of irreducible Markov chains on n states. We assume that the chain is rapidly mixing and with a stationary measure with no entry being either too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state x—for the chain started at stationarity—up to a small multiplicative correction. While the proof by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob’s transform of the chain on the complement of the state x.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44179220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The lattice Green function, i.e., the resolvent of the discrete Laplace operator, is fundamental in probability theory and mathematical physics. We derive its long-distance behaviour via a detailed analysis of an integral representation involving modified Bessel functions. Our emphasis is on the decay of the massive lattice Green function in the vicinity of the massless (critical) case, and the recovery of Euclidean isotropy in the massless limit. This provides a prototype for the expected but unproven long-distance behaviour of near-critical two-point functions in statistical mechanical models such as percolation, the Ising model, and the self-avoiding walk above their upper critical dimensions.
{"title":"Asymptotic behaviour of the lattice Green function","authors":"Emmanuel Michta, G. Slade","doi":"10.30757/alea.v19-38","DOIUrl":"https://doi.org/10.30757/alea.v19-38","url":null,"abstract":"The lattice Green function, i.e., the resolvent of the discrete Laplace operator, is fundamental in probability theory and mathematical physics. We derive its long-distance behaviour via a detailed analysis of an integral representation involving modified Bessel functions. Our emphasis is on the decay of the massive lattice Green function in the vicinity of the massless (critical) case, and the recovery of Euclidean isotropy in the massless limit. This provides a prototype for the expected but unproven long-distance behaviour of near-critical two-point functions in statistical mechanical models such as percolation, the Ising model, and the self-avoiding walk above their upper critical dimensions.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45392594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we prove that k-dimensional spherical integrals are asymptotically equivalent to the product of 1-dimensional spherical integrals. This allows us to generalize several large deviations principles in random matrix theory known before only in a one-dimensional case. As examples, we study the universality of the large deviations for k extreme eigenvalues of Wigner matrices (resp. Wishart matrices, resp. matrices with general variance profiles) with sharp sub-Gaussian entries, as well as large deviations principles for extreme eigenvalues of Gaussian Wigner and Wishart matrices with a finite dimensional perturbation.
{"title":"Asymptotics of k dimensional spherical integrals and applications","authors":"A. Guionnet, Jonathan Husson","doi":"10.30757/alea.v19-30","DOIUrl":"https://doi.org/10.30757/alea.v19-30","url":null,"abstract":"In this article, we prove that k-dimensional spherical integrals are asymptotically equivalent to the product of 1-dimensional spherical integrals. This allows us to generalize several large deviations principles in random matrix theory known before only in a one-dimensional case. As examples, we study the universality of the large deviations for k extreme eigenvalues of Wigner matrices (resp. Wishart matrices, resp. matrices with general variance profiles) with sharp sub-Gaussian entries, as well as large deviations principles for extreme eigenvalues of Gaussian Wigner and Wishart matrices with a finite dimensional perturbation.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41878689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using coupling techniques based on Stein’s method for probability approximation, we revisit classical variance bounding inequalities of Chernoff, Cacoullos, Chen and Klaassen. Our bounds are immediate in any context wherein a Stein identity is available. After providing illustrative examples for a Gaussian and a Gumbel target distribution, our main contributions are new variance bounds in settings where the underlying density function is unknown or intractable. Applications include bounds for analysis of the posterior in Bayesian statistics, bounds for asymptotically Gaussian random variables using zero-biased couplings, and bounds for random variables which are New Better (Worse) than Used in Expectation.
{"title":"Some simple variance bounds from Stein’s method","authors":"Fraser Daly, Fatemeh Ghaderinezhad, Christophe Ley, Yvik Swan","doi":"10.30757/alea.v18-69","DOIUrl":"https://doi.org/10.30757/alea.v18-69","url":null,"abstract":"Using coupling techniques based on Stein’s method for probability approximation, we revisit classical variance bounding inequalities of Chernoff, Cacoullos, Chen and Klaassen. Our bounds are immediate in any context wherein a Stein identity is available. After providing illustrative examples for a Gaussian and a Gumbel target distribution, our main contributions are new variance bounds in settings where the underlying density function is unknown or intractable. Applications include bounds for analysis of the posterior in Bayesian statistics, bounds for asymptotically Gaussian random variables using zero-biased couplings, and bounds for random variables which are New Better (Worse) than Used in Expectation.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Excursion theory is revisited on the ground of Itô–McKean diffusions. There are raised questions about symmetries, knock-in processes, excursion local time and the non-linear version of the master formula of excursions. The questions are answered due to introducing the counting excursion technique. The technique is a synthesis of straddling at time approach, the classical, potential in spirit approach, and the theory of convolution algebra of locally integrable functions, generalized later in this work for the convolutions of σ–finite measures. Some examples are presented, including the famous problem of expressing the density of first hitting time of Ornstein-Uhlenbeck process in terms of elementary functions.
{"title":"Counting excursions: symmetries, knock-ins and\u0000non-linear formula for Itô--McKean diffusions","authors":"Maciej Wiśniewolski","doi":"10.30757/ALEA.V18-16","DOIUrl":"https://doi.org/10.30757/ALEA.V18-16","url":null,"abstract":"Excursion theory is revisited on the ground of Itô–McKean diffusions. There are raised questions about symmetries, knock-in processes, excursion local time and the non-linear version of the master formula of excursions. The questions are answered due to introducing the counting excursion technique. The technique is a synthesis of straddling at time approach, the classical, potential in spirit approach, and the theory of convolution algebra of locally integrable functions, generalized later in this work for the convolutions of σ–finite measures. Some examples are presented, including the famous problem of expressing the density of first hitting time of Ornstein-Uhlenbeck process in terms of elementary functions.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"29 1","pages":"349"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84340018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove sharp two-sided estimates on the tail probability of the first hitting time of bounded interval as well as its asymptotic behaviour for general non-symmetric processes which satisfy an integral condition ∫ ∞ 0 dξ 1 + Reψ(ξ) < ∞. To this end, we first prove and then apply the global scale invariant Harnack inequality. Results are obtained under certain conditions on the characteristic exponent. We provide a wide class of Lévy processs which satisfy these assumptions.
{"title":"Hitting probabilities for Lévy processes\u0000on the real line","authors":"T. Grzywny, Łukasz Leżaj, Maciej Miśta","doi":"10.30757/ALEA.V18-27","DOIUrl":"https://doi.org/10.30757/ALEA.V18-27","url":null,"abstract":"We prove sharp two-sided estimates on the tail probability of the first hitting time of bounded interval as well as its asymptotic behaviour for general non-symmetric processes which satisfy an integral condition ∫ ∞ 0 dξ 1 + Reψ(ξ) < ∞. To this end, we first prove and then apply the global scale invariant Harnack inequality. Results are obtained under certain conditions on the characteristic exponent. We provide a wide class of Lévy processs which satisfy these assumptions.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":"727"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82307341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a doubly reflected backward stochastic differential equations with jumps and two completely separated optional barriers in a general filtration that supports a one-dimensional Brownian motion and an independent Poisson random measure. We suppose that the barriers have trajectories with left and right finite limits. We provide the existence and uniqueness result when the coefficient is stochastic Lipschitz by using a penalization method.
{"title":"BSDEs with jumps and two completely separated\u0000irregular barriers in a general filtration","authors":"M. Marzougue, M. Otmani","doi":"10.30757/ALEA.V18-28","DOIUrl":"https://doi.org/10.30757/ALEA.V18-28","url":null,"abstract":"We consider a doubly reflected backward stochastic differential equations with jumps and two completely separated optional barriers in a general filtration that supports a one-dimensional Brownian motion and an independent Poisson random measure. We suppose that the barriers have trajectories with left and right finite limits. We provide the existence and uniqueness result when the coefficient is stochastic Lipschitz by using a penalization method.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"18 1","pages":"761"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75474549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The investigation of the spectra of finite Markov generators was first motivated by the quest of quantitative bounds on the convergence to equilibrium of the corresponding processes, see for instance the reference book Levin et al. (2009). There is also a classification reason: what are the possible spectra of Markov generators?, and for such a given spectrum, is there a simple representative Markov process? This structural question is related to the previous motivation, as ergodic finite isospectral Markov generators can be intertwined and under certain circumstances this relation enables good transfers of information about the speed of convergence, see e.g. Miclo (2018); Miclo and Patie (2021). Our goal here goes in this general direction, by providing a simple family of Markov generators for real spectra with geometric multiplicity 1.
对有限马尔可夫生成器谱的研究最初是由对相应过程收敛到平衡的定量界限的探索所激发的，例如参见参考书Levin et al.(2009)。还有一个分类原因:马尔可夫发生器的可能谱是什么?，对于这样一个给定的频谱，是否存在一个简单的具有代表性的马尔可夫过程?这个结构问题与前面的动机有关，因为遍历有限等谱马尔可夫生成器可以交织在一起，并且在某些情况下，这种关系能够很好地传递关于收敛速度的信息，例如Miclo (2018);Miclo and Patie(2021)。我们的目标是沿着这个大致的方向，通过提供一组简单的马尔可夫发生器，用于具有几何多重性1的实谱。
{"title":"On A Family of Isospectral Pure-Birth Processes","authors":"L. Miclo, Chi Zhang","doi":"10.30757/alea.v18-65","DOIUrl":"https://doi.org/10.30757/alea.v18-65","url":null,"abstract":"The investigation of the spectra of finite Markov generators was first motivated by the quest of quantitative bounds on the convergence to equilibrium of the corresponding processes, see for instance the reference book Levin et al. (2009). There is also a classification reason: what are the possible spectra of Markov generators?, and for such a given spectrum, is there a simple representative Markov process? This structural question is related to the previous motivation, as ergodic finite isospectral Markov generators can be intertwined and under certain circumstances this relation enables good transfers of information about the speed of convergence, see e.g. Miclo (2018); Miclo and Patie (2021). Our goal here goes in this general direction, by providing a simple family of Markov generators for real spectra with geometric multiplicity 1.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For exchangeable coalescents with dust the rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate is expressed in terms of a certain Bernstein function. The proofs are based on Taylor expansions of the infinitesimal generators and semigroups and involve a particular concentration inequality arising in the context of Karlin’s infinite urn model.
{"title":"The rate of convergence of the block counting process\u0000of exchangeable coalescents with dust","authors":"M. Möhle","doi":"10.30757/alea.v18-44","DOIUrl":"https://doi.org/10.30757/alea.v18-44","url":null,"abstract":"For exchangeable coalescents with dust the rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate is expressed in terms of a certain Bernstein function. The proofs are based on Taylor expansions of the infinitesimal generators and semigroups and involve a particular concentration inequality arising in the context of Karlin’s infinite urn model.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69590917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：