{"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}
{"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}
{"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}
{"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}
{"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}
{"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}