A new look at independence michel talagrand abstract. The law of the iterated logarithm for a triangular array of empirical processes arcones, miguel, electronic journal of probability, 1997. Informally, the tails of a subgaussian distribution are dominated by i. The reader is assumed to be familiar with classical methods from the theory of processes e.
It turns out that many global properties of banach spaces can be learned from properties of. For the purposes of these notes, all vector spaces are assumed to be over the real numbers. Statistical mechanics, spin glasses, gibbs measures. This volume collects selected papers from the 8th high dimensional probability meeting held at casa matematica oaxaca cmo, mexico. Refinement of convergence rate for the strong law of large numbers in banach space november 2014 stochastics an international journal of probability and stochastic processes 866. Banach spaces originally grew out of the study of function spaces by hilbert, frechet, and riesz earlier in the century. Four talagrand inequalities under the same umbrella. From this analytical point of view, emphasis is put on inequalities in nite dimension as well as on the fundamental gaussian measurable structure consisting of the product measure on irin when.
Concentration of measure inequalities in information theory, communications and coding. This concept now plays an important role in the local theory of banach spaces, and the dominant role in probability in banach space. Normed and banach spaces in this chapter we introduce the basic setting of functional analysis, in the form of normed spaces and bounded linear operators. This twovolume text provides a complete overview of the theory of banach spaces, emphasising its interplay with classical and harmonic analysis particularly sidon sets and probability. Empirical process methods in probability in banach spaces. Toward a general law of the iterated logarithm in banach space einmahl, uwe, the annals of probability, 1993. Ams theory of probability and mathematical statistics. On the other hand, any consistent lifting for an hyperstonian space is not strong, see strauss et al. This is the file of a preliminary version, and the references are missing.
Talagrand, probability in banach spaces, springer, berlin, 1991. A prime example of space where concentration of measure hoids is the euclidean. Written for graduate students, these books are also a valuable reference for researchers in analysis. Michel talagrand isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of probability in banach spaces. Michel talagrand new isoperimetric inequalities and random process techniques have recently appeared at the basis of the modern understanding of probability in banach spaces. The meaning of the word most is made rigorous by isoperimetrictype inequalities that bound the measure of the. The concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables. On page 5, of this book, there is a list of previous conferences in probability in banach spaces, including the other eight international conferences. For arbitrary banach spaces, i wouldnt count on it.
The authors give a full exposition of all results, as well as numerous exercises and comments to complement the. Random variables in banach spaces in this lecture we take up the study of random variables with values in a banach space e. Volume 1 covers the basics of banach space theory, operatory theory in banach spaces, harmonic analysis and probability. A good understanding about the disciplines involved in this field can be obtained from the recent book, probability in banach spaces, springerverlag, by m. Pdf characterization of gaussian semigroups on separable. This cited by count includes citations to the following articles in scholar. Probability in banach spaces michel ledoux, michel. Pdf probability distributions on banach spaces mathematics. We prove a strong convergence theorem for banach space valued random variables. A large number of applications are given, in particular in percolation, geometric probability, probability in banach spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools. Probability in banach spaces isoperimetry and processes. This book gives an excellent, almost complete account of the whole subject of probability in banach spaces, a branch of probability theory that has undergone vigorous development during the last thirty years. Complete probability space an overview sciencedirect topics. A large number of applications are given, in particular in percolation, geometric probability, probability in banach spaces, to demonstrate in concrete situations the extremely wide range of.
The ledoux talagrand book has been reprinted in the affordable series classics in mathematics. Kolmogorovchentsov criteria, the garsiarodemichrumsey lemma or martingale methods, but as far. Some problems of probability theory in banach space ivan matsak taras shevchenko national university of kyiv, kyiv, ukraine the talk will be devoted to the law of large numbers, the central limit theorem, the law of the iterated logarithm and their applications in banach space theory. Michel pierre talagrand born 15 february 1952 is a french mathematician. Concentration around the mean for maxima of empirical processes klein, t. Banach spaces are named after the polish mathematician stefan banach, who introduced this concept and studied it systematically in 19201922 along with hans hahn and eduard helly. The results ii to iv are analogs of results proved by moran 1969 for strongly measure compact spaces. Characterization of the law of the iterated logarithm in banach spaces. The proof uses tools of the theory of gaussian random vectors as well as by now classical arguments of probability in banach spaces. Improved convergence rates for tail probabilities for sums of i. A banach space e under its weak topology is lifting compact if and only if every evalued scalarly measurable function is scalarly equivalent to a bochner measurable function by bellow 1980, section 6, remark 2. Hahn tufts university this is a book about the importance of inequalities and the power of isoperimetric methods.
Montgomerysmith department of mathematics, university of missouri. Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of probability in banach spaces. Banach, spaces and the process of completion of a normed space to a banach space. His research interests centre on probability, random matrices, logarithmic sobolev inequalities, probability in banach spaces. Talagrand concentration inequalities for spde 3 where. Gaussian measures and processes, probability in banach spaces, convexity. Complete metric spaces notions such as convergent sequence and cauchy sequence make sense for any metric space. The authors also provide an annex devoted to compact abelian groups. Introduction to the nonasymptotic analysis of random matrices. Concentration of measure and isoperimetric inequalities in. Concentration of measure and isoperimetric inequalities in product.
Improved convergence rates for tail probabilities for sums. Some problems of probability theory in banach space ivan. Throughout, f will denote either the real line r or the complex plane c. N of probability spaces has measure at least one half, most of the points of. The concentration of measure phenomenon in product spaces roughly states that, if a set a in a product. In section 2 we recall talagrand concentration inequalities and state our main results.
Pdf refinement of convergence rate for the strong law of. The notes aim at an audience that feels more at ease in analysis than in probability theory. High dimensional probability hdp is an area of mathematics that includes the study of probability distributions and limit theorems in infinitedimensional spaces such as hilbert spaces and banach spaces. The book will also be an invaluable reference volume for researchers in analysis. His approach consists in using results on gaussian semigroups to investigate the associated stationary problems. Isoperimetry and processes michel ledoux, michel talagrand auth. Probability in banach spaces isoperimetry and processes michel.
Lecture notes settheoretic methods in banach spaces piotr koszmider given at the institute of mathematics of the university of wroc law as a part of the series set theory applied within the programme of. Random matrices, orthogonal polynomials, free probability. Volumes of convex bodies and banach space geometry tomczak, jaegerman. The distribution of a random variable in a banach space xwill be a probability measure on x. Probability in banach spaces, 9 jorgen hoffmannjorgensen. Probability measures on metric spaces onno van gaans.
Isoperimetry and processes in probability in a banach space, springerverlag to appear. Formally, the probability distribution of a random variable x is called subgaussian if there are positive constants c, v such that for every t 0. Talagrand 1982, 1988 introduced them, proved their existence and has shown that these objects give a simple and efficient way to deal with measurability problems for a class of stochastic process. Based on these tools, the book presents a complete. Thus, in the modern guise, functional analysis is the study of banach spaces and bounded linear opera. Namely, let e be a separable banach space with a nondegenerate centered gaussian measure. His thesis was directed by gustave choquet and his interests revolve around the theory of stochastic processes and probability in banach spaces, as.
We proceed to a systematic exploration of this phenomenon. Michel talagrand has held a research position with the cnrs since 1974. Pdf empirical process methods in probability in banach spaces. In lectures i proceed to the next chapter, on lebesgue. Actually, most of the tools and inspiration come from analysis rather than probability. Folge a series of modern surveys in mathematics 1st edition by michel ledoux, michel talagrand hardcover, 480 pages, published 1991. Based on these tools, the book presents a complete treatment of the main aspects of probability in banach spaces integrability and limit theorems. Of course, this means that we have to start caring about. This twovolume text provides a complete overview of the theory of banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of probability in banach spaces integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes and of some of their links to geometry of banach spaces via the type and cotype properties. Michel ledoux, michel talagrand, probability in banach spaces. See the book lt2, and subsequent work such as t6, t7. A strong convergence theorem for banach space valued random variables article pdf available in the annals of probability 45 october 1976 with 31 reads how we measure reads. Banach spaces these notes provide an introduction to banach spaces, which are complete normed vector spaces.
Isoperimetry and integrability of the sum of independent banach space valued random variables talagrand, michel, the annals of probability, 1989. The ones marked may be different from the article in the profile. In probability theory, a subgaussian distribution is a probability distribution with strong tail decay. Improved rademacher symmetrization through a wasserstein. Concentration inequalities by boucheron, lugosi, massart. Based on these tools, the book presents a complete treatment of the main aspects of probability in banach spaces integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes and of some of their links to geometry of banach spaces via the type and. Geometry of markov operators, riemannian geometry, path and loop spaces. Pdf a strong convergence theorem for banach space valued. Ledoux probability in banach spaces springer, berlin, 1991. Still springerverlag has most generously agreed that i distribute the file below. Isoperimetric, measure concentration and random process.