{"title":"Geometry of information structures, strategic measures and associated stochastic control topologies","authors":"Naci Saldi, S. Yüksel","doi":"10.1214/20-ps356","DOIUrl":"https://doi.org/10.1214/20-ps356","url":null,"abstract":"","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42251302","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 describe some results on approximation of convex bodies by polytopes. Best and random approximations are considered and compared. The geometric quantities related to the convex body, that appear naturally in such approximation questions are the affine surface areas. Those, and their relation to floating bodies, will be discussed as well. In this survey we give only a very selective collection of results on ap- proximation by polytopes.
{"title":"Floating bodies and approximation of convex bodies by polytopes","authors":"E. Werner","doi":"10.1214/22-ps5","DOIUrl":"https://doi.org/10.1214/22-ps5","url":null,"abstract":": We describe some results on approximation of convex bodies by polytopes. Best and random approximations are considered and compared. The geometric quantities related to the convex body, that appear naturally in such approximation questions are the affine surface areas. Those, and their relation to floating bodies, will be discussed as well. In this survey we give only a very selective collection of results on ap- proximation by polytopes.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47769416","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 study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a solution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the boundary of the domain combined with hypoellipticity of the operators associated to the SDE. The main general results are then applied to deduce properties of the associated Green’s functions and to obtain a generalization of Bony’s Harnack inequality. We moreover revisit the transience and recurrence dichotomy for hypoelliptic diffusions and its relationship to invariant measures.
{"title":"The method of stochastic characteristics for linear second-order hypoelliptic equations","authors":"J. Foldes, David P. Herzog","doi":"10.1214/22-ps11","DOIUrl":"https://doi.org/10.1214/22-ps11","url":null,"abstract":"We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a solution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the boundary of the domain combined with hypoellipticity of the operators associated to the SDE. The main general results are then applied to deduce properties of the associated Green’s functions and to obtain a generalization of Bony’s Harnack inequality. We moreover revisit the transience and recurrence dichotomy for hypoelliptic diffusions and its relationship to invariant measures.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49560800","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}
In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp. rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp. its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix. ∗Corresponding author. Division of Mathematics and Physics, Mälardalen University, Box 883, 721 23 Västerås, Sweden. E-mail: anatoliy.malyarenko@mdh.se. †Department of Mechanical Science & Engineering, also Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL, 61801-2906 USA. E-mail: martinos@illinois.edu. 1 ar X iv :2 11 2. 04 82 6v 1 [ m at h. PR ] 9 D ec 2 02 1
{"title":"Tensor- and spinor-valued random fields with applications to continuum physics and cosmology","authors":"A. Malyarenko, M. Ostoja-Starzewski","doi":"10.1214/22-ps12","DOIUrl":"https://doi.org/10.1214/22-ps12","url":null,"abstract":"In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp. rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp. its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix. ∗Corresponding author. Division of Mathematics and Physics, Mälardalen University, Box 883, 721 23 Västerås, Sweden. E-mail: anatoliy.malyarenko@mdh.se. †Department of Mechanical Science & Engineering, also Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL, 61801-2906 USA. E-mail: martinos@illinois.edu. 1 ar X iv :2 11 2. 04 82 6v 1 [ m at h. PR ] 9 D ec 2 02 1","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44580194","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 study the recurrence of homogeneous and oscillating random walks on the integers, simplifying former works of Spitzer and Kemperman, respectively. We add general remarks and discuss some links with renewal theory.
{"title":"On homogeneous and oscillating random walks on the integers","authors":"Julien Br'emont","doi":"10.1214/23-ps13","DOIUrl":"https://doi.org/10.1214/23-ps13","url":null,"abstract":"We study the recurrence of homogeneous and oscillating random walks on the integers, simplifying former works of Spitzer and Kemperman, respectively. We add general remarks and discuss some links with renewal theory.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49139792","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}
This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis–Żelazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.
{"title":"Infinite convolutions of probability measures on Polish semigroups","authors":"K. Yano","doi":"10.1214/22-ps6","DOIUrl":"https://doi.org/10.1214/22-ps6","url":null,"abstract":"This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis–Żelazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43970724","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 present a version of the RSK correspondence based on the Pitman transform and geometric considerations. This version unifies ordinary RSK, dual RSK and continuous RSK. We show that this version is both a bijection and an isometry, two crucial properties for taking limits of last passage percolation models. We use the bijective property to give a non-computational proof that dual RSK maps Bernoulli walks to nonintersecting Bernoulli walks.
{"title":"RSK in last passage percolation: a unified approach","authors":"Duncan Dauvergne, M. Nica, B'alint Vir'ag","doi":"10.1214/22-PS4","DOIUrl":"https://doi.org/10.1214/22-PS4","url":null,"abstract":": We present a version of the RSK correspondence based on the Pitman transform and geometric considerations. This version unifies ordinary RSK, dual RSK and continuous RSK. We show that this version is both a bijection and an isometry, two crucial properties for taking limits of last passage percolation models. We use the bijective property to give a non-computational proof that dual RSK maps Bernoulli walks to nonintersecting Bernoulli walks.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48589565","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}
These notes survey the first results on large deviations of Schramm-Loewner evolutions (SLE) with emphasis on interrelations between rate functions and applications to complex analysis. More precisely, we describe the large deviations of SLE$_kappa$ when the $kappa$ parameter goes to zero in the chordal and multichordal case and to infinity in the radial case. The rate functions, namely Loewner and Loewner-Kufarev energies, are closely related to the Weil-Petersson class of quasicircles and real rational functions.
{"title":"Large deviations of Schramm-Loewner evolutions: A survey","authors":"Yilin Wang","doi":"10.1214/22-ps9","DOIUrl":"https://doi.org/10.1214/22-ps9","url":null,"abstract":"These notes survey the first results on large deviations of Schramm-Loewner evolutions (SLE) with emphasis on interrelations between rate functions and applications to complex analysis. More precisely, we describe the large deviations of SLE$_kappa$ when the $kappa$ parameter goes to zero in the chordal and multichordal case and to infinity in the radial case. The rate functions, namely Loewner and Loewner-Kufarev energies, are closely related to the Weil-Petersson class of quasicircles and real rational functions.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44623091","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}
The survey is dedicated to a celebrated series of quantitave results, developed by the Lithuanian school of probability, on the normal approximation for a real-valued random variable. The key ingredient is a bound on cumulants of the type |κj(X)| ≤ j!1+γ/∆j−2, which is weaker than Cramér’s condition of finite exponential moments. We give a self-contained proof of some of the “main lemmas” in a book by Saulis and Statulevičius (1989), and an accessible introduction to the Cramér-Petrov series. In addition, we explain relations with heavy-tailed Weibull variables, moderate deviations, and mod-phi convergence. We discuss some methods for bounding cumulants such as summability of mixed cumulants and dependency graphs, and briefly review a few recent applications of the method of cumulants for the normal approximation. Mathematics Subject Classification 2020: 60F05; 60F10; 60G70; 60K35.
{"title":"The method of cumulants for the normal approximation","authors":"Hanna Doring, S. Jansen, K. Schubert","doi":"10.1214/22-ps7","DOIUrl":"https://doi.org/10.1214/22-ps7","url":null,"abstract":"The survey is dedicated to a celebrated series of quantitave results, developed by the Lithuanian school of probability, on the normal approximation for a real-valued random variable. The key ingredient is a bound on cumulants of the type |κj(X)| ≤ j!1+γ/∆j−2, which is weaker than Cramér’s condition of finite exponential moments. We give a self-contained proof of some of the “main lemmas” in a book by Saulis and Statulevičius (1989), and an accessible introduction to the Cramér-Petrov series. In addition, we explain relations with heavy-tailed Weibull variables, moderate deviations, and mod-phi convergence. We discuss some methods for bounding cumulants such as summability of mixed cumulants and dependency graphs, and briefly review a few recent applications of the method of cumulants for the normal approximation. Mathematics Subject Classification 2020: 60F05; 60F10; 60G70; 60K35.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46472388","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}
This is a survey of results on universal algorithms for classification and prediction of stationary processes. The classification problems include discovering the order of a k-step Markov chain, determining memory words in finitarily Markovian processes and estimating the entropy of an unknown process. The prediction problems cover both discrete and real valued processes in a variety of situations. Both the forward and the backward prediction problems are discussed with the emphasis being on pointwise results. This survey is just a teaser. The purpose is merely to call attention to results on classification and prediction. We will refer the interested reader to the sources. Throughout the paper we will give illuminating examples. AMS 2000 subject classifications: Primary 60G25, 60G10.
{"title":"On universal algorithms for classifying and predicting stationary processes","authors":"G. Morvai, B. Weiss","doi":"10.1214/20-PS345","DOIUrl":"https://doi.org/10.1214/20-PS345","url":null,"abstract":"This is a survey of results on universal algorithms for classification and prediction of stationary processes. The classification problems include discovering the order of a k-step Markov chain, determining memory words in finitarily Markovian processes and estimating the entropy of an unknown process. The prediction problems cover both discrete and real valued processes in a variety of situations. Both the forward and the backward prediction problems are discussed with the emphasis being on pointwise results. This survey is just a teaser. The purpose is merely to call attention to results on classification and prediction. We will refer the interested reader to the sources. Throughout the paper we will give illuminating examples. AMS 2000 subject classifications: Primary 60G25, 60G10.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"18 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66085057","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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