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An introduction to measure and probability / John Christopher Taylor / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1997)
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Titre : An Introduction to Measure and Probability Type de document : document électronique Auteurs : John Christopher Taylor (1936-....) ; SpringerLink (Online service) Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1997 Importance : XVII, 324 p. 4 illus Présentation : online resource ISBN/ISSN/EAN : 978-1-4612-0659-0 Langues : Anglais (eng) Tags : Mathematics Functions of real variables Probabilities Probability Theory and Stochastic Processes Real Functions Résumé : Assuming only calculus and linear algebra, this book introduces the reader in a technically complete way to measure theory and probability, discrete martingales, and weak convergence. It is self- contained and rigorous with a tutorial approach that leads the reader to develop basic skills in analysis and probability. While the original goal was to bring discrete martingale theory to a wide readership, it has been extended so that the book also covers the basic topics of measure theory as well as giving an introduction to the Central Limit Theory and weak convergence. Students of pure mathematics and statistics can expect to acquire a sound introduction to basic measure theory and probability. A reader with a background in finance, business, or engineering should be able to acquire a technical understanding of discrete martingales in the equivalent of one semester. J. C. Taylor is a Professor in the Department of Mathematics and Statistics at McGill University in Montreal. He is the author of numerous articles on potential theory, both probabilistic and analytic, and is particularly interested in the potential theory of symmetric spaces Note de contenu : I. Probability Spaces -- 1. Introduction to ? -- 2. What is a probability space? Motivation -- 3. Definition of a probability space -- 4. Construction of a probability from a distribution function -- 5. Additional exercises* -- II. Integration -- 1. Integration on a probability space -- 2. Lebesgue measure on ? and Lebesgue integration -- 3. The Riemann integral and the Lebesgue integral -- 4. Probability density functions -- 5. Infinite series again -- 6. Differentiation under the integral sign -- 7. Signed measures and the Radon-Nikodym theorem* -- 8. Signed measures on ? and functions of bounded variation* -- 9. Additional exercises* -- III. Independence and Product Measures -- 1. Random vectors and Borel sets in ?n -- 2. Independence -- 3. Product measures -- 4. Infinite products -- 5. Some remarks on Markov chains* -- 6. Additional exercises* -- IV. Convergence of Random Variables and Measurable Functions -- 1. Norms for random variables and measurable functions -- 2. Continuous functions and Lp* -- 3. Pointwise convergence and convergence in measure or probability -- 4. Kolmogorov?s inequality and the strong law of large numbers -- 5. Uniform integrability and truncation* -- 6. Differentiation: the Hardy?Littlewood maximal function* -- 7. Additional exercises* -- V. Conditional Expectation and an Introduction to Martingales -- 1. Conditional expectation and Hilbert space -- 2. Conditional expectation -- 3. Sufficient statistics* -- 4. Martingales -- 5. An introduction to martingale convergence -- 6. The three-series theorem and the Doob decomposition -- 7. The martingale convergence theorem -- VI. An Introduction to Weak Convergence -- 1. Motivation: empirical distributions -- 2. Weak convergence of probabilities: equivalent formulations -- 3. Weak convergence of random variables -- 4. Empirical distributions again: the Glivenko?Cantelli theorem -- 5. The characteristic function -- 6. Uniqueness and inversion of the characteristic function -- 7. The central limit theorem -- 8. Additional exercises* -- 9. Appendix* Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=126942 An Introduction to Measure and Probability [document électronique] / John Christopher Taylor (1936-....) ; SpringerLink (Online service) . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1997 . - XVII, 324 p. 4 illus : online resource.
ISBN : 978-1-4612-0659-0
Langues : Anglais (eng)
Tags : Mathematics Functions of real variables Probabilities Probability Theory and Stochastic Processes Real Functions Résumé : Assuming only calculus and linear algebra, this book introduces the reader in a technically complete way to measure theory and probability, discrete martingales, and weak convergence. It is self- contained and rigorous with a tutorial approach that leads the reader to develop basic skills in analysis and probability. While the original goal was to bring discrete martingale theory to a wide readership, it has been extended so that the book also covers the basic topics of measure theory as well as giving an introduction to the Central Limit Theory and weak convergence. Students of pure mathematics and statistics can expect to acquire a sound introduction to basic measure theory and probability. A reader with a background in finance, business, or engineering should be able to acquire a technical understanding of discrete martingales in the equivalent of one semester. J. C. Taylor is a Professor in the Department of Mathematics and Statistics at McGill University in Montreal. He is the author of numerous articles on potential theory, both probabilistic and analytic, and is particularly interested in the potential theory of symmetric spaces Note de contenu : I. Probability Spaces -- 1. Introduction to ? -- 2. What is a probability space? Motivation -- 3. Definition of a probability space -- 4. Construction of a probability from a distribution function -- 5. Additional exercises* -- II. Integration -- 1. Integration on a probability space -- 2. Lebesgue measure on ? and Lebesgue integration -- 3. The Riemann integral and the Lebesgue integral -- 4. Probability density functions -- 5. Infinite series again -- 6. Differentiation under the integral sign -- 7. Signed measures and the Radon-Nikodym theorem* -- 8. Signed measures on ? and functions of bounded variation* -- 9. Additional exercises* -- III. Independence and Product Measures -- 1. Random vectors and Borel sets in ?n -- 2. Independence -- 3. Product measures -- 4. Infinite products -- 5. Some remarks on Markov chains* -- 6. Additional exercises* -- IV. Convergence of Random Variables and Measurable Functions -- 1. Norms for random variables and measurable functions -- 2. Continuous functions and Lp* -- 3. Pointwise convergence and convergence in measure or probability -- 4. Kolmogorov?s inequality and the strong law of large numbers -- 5. Uniform integrability and truncation* -- 6. Differentiation: the Hardy?Littlewood maximal function* -- 7. Additional exercises* -- V. Conditional Expectation and an Introduction to Martingales -- 1. Conditional expectation and Hilbert space -- 2. Conditional expectation -- 3. Sufficient statistics* -- 4. Martingales -- 5. An introduction to martingale convergence -- 6. The three-series theorem and the Doob decomposition -- 7. The martingale convergence theorem -- VI. An Introduction to Weak Convergence -- 1. Motivation: empirical distributions -- 2. Weak convergence of probabilities: equivalent formulations -- 3. Weak convergence of random variables -- 4. Empirical distributions again: the Glivenko?Cantelli theorem -- 5. The characteristic function -- 6. Uniqueness and inversion of the characteristic function -- 7. The central limit theorem -- 8. Additional exercises* -- 9. Appendix* Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=126942 Compactification of Symmetric Spaces / Yves Guivarc'h / Bâle (CHE) ; Boston, MA : Birkhäuser (1998)
Titre : Compactification of Symmetric Spaces Type de document : document électronique Auteurs : Yves Guivarc'h ; Lizhen Ji ; SpringerLink (Online service) ; John Christopher Taylor (1936-....) Editeur : Bâle (CHE) ; Boston, MA : Birkhäuser Année de publication : 1998 Collection : Progress in Mathematics, ISSN 0743-1643 num. 156 Importance : XIV, 284 p Présentation : online resource ISBN/ISSN/EAN : 978-1-4612-2452-5 Langues : Anglais (eng) Tags : Mathematics Geometry Topology Résumé : The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view. Key features: * definition and detailed analysis of the Martin compactifications * new geometric Compactification, defined in terms of the Tits building, that coincides with the Martin Compactification at the bottom of the positive spectrum. * geometric, non-inductive, description of the Karpelevic Compactification * study of the well-know isomorphism between the Satake compactifications and the Furstenberg compactifications * systematic and clear progression of topics from geometry to analysis, and finally to random walks The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students Note de contenu : I. Introduction -- Statement of the main new results -- Characterizations of the compactification $${\bar X^{SF}}$$ -- The Karpelevi? compactification $${\bar X^K}$$ -- Fibers of maps between the compactifications -- Application to Brownian motion -- Eigenfunctions and Martin?s method -- Methods of proof -- Open problems -- Conventions -- Study guide -- II. Subalgebras and parabolic subgroups -- The Iwasawa and Cartan decompositions -- Parabolic subgroups -- Subsets of ? and Lie subalgebras -- The Langlands decomposition of PI and the symmetric space XI -- Bruhat decompositions -- III. Geometrical constructions of compactifications -- The conic compactification $${\bar X^c}$$ -- The conical decomposition of a and the Weyl group -- Parabolic subgroups and stabilizers of the points in X(?) -- Flats through the base point and Proposition 3.8 -- The Tits building ?(G) of G and its geometrical realization ?(X) -- The polyhedral compactification of a flat -- The dual cell complex ?*(X) -- The dual cell compactification X ? ?*(X) -- IV. The Satake?Furstenberg compactifications -- Finite dimensional representations -- Weights and highest weights -- Representation and parabolic subgroups -- Satake compactifications -- Furstenberg compactifications -- V. The Karpelevi? compactification -- The Karpelevi? compactification -- Convergence in the Karpelevi? topology restricted to a flat -- The Karpelevi? compactification of a -- The Karpelevi? topology is compact -- The relation between the Karpelevi? compactification, conical and dual cell compactifications -- A characterization of the Karpelevi? compactification -- VI. Martin compactifications -- The Martin compactification -- Convergence of Brownian motion -- Extension of the group action to the Martin compactification -- The Martin compactification for a random walk -- VII. The Martin compactification X ? ?X(?0) -- The Laplacian in horocyclic coordinates -- Generalized horocyclic coordinates and the Laplacian -- Computation of the limit functions: reduction -- The limit of a CI-canonical sequence -- Classification of limit functions and the topology of X ? ?X(?0) -- VIII. The Martin compactification X ? ?X(?) -- The case of X = SL(n, ?)/SU(n) for ? < ?0 -- Computation of the limit functions for a general semisimple group -- Determination of the Martin compactification -- Bounded harmonic functions on X -- An application to convergence of Brownian motion -- IX. An intrinsic approach to the boundaries of X -- The space of closed subgroups -- Limit groups -- Limits of group spheres -- Parabolic subgroups and boundary theory -- The maximal Furstenberg compactification -- X. Compactification via the ground state -- The twisted action -- Compactification of X via the ground state -- XI. Harnack inequality, Martin?s method and the positive spectrum for random walks -- Basic notations -- Cones with compact bases and the Harnack inequality -- Martin?s method for a random walk -- The positive spectrum of a random walk -- The fixed line property -- Formulas for r(p),r0(p) -- Outline of the following chapters -- XII. The Furstenberg boundary and bounded harmonic functions -- Basic notations -- The mean-value property -- Harmonic functions and the mean-value property -- Convergence theorems for harmonic functions -- The Poisson formula for random walks -- XIII. Integral representation of positive eigenfunctions of convolution operators -- The main result of this chapter -- An extension of the main result -- Analytic determination of the minimal eigenfunctions of the Laplacian -- The Busemann cocycle and a geometrical determination of the minimal eigenfunctions of the Laplacian -- Minimal eigenfunctions for random walks -- XIV. Random walks and ground state properties -- Basic definitions and properties -- Convolution -- Spherical functions and minimal eigenfunctions -- Ground state properties -- Random walks, eigenfunctions of the Laplacian and X ? ?X(?0) -- The Martin compactification of X determined by a random walk -- An application to parabolic subgroups -- XV. Extension to semisimple algebraic groups defined over a local field -- Some notations and fundamental properties -- Extension of the main results of Chapters XII, XIII, XIV -- Appendix A -- Appendix B -- List of symbols Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=118670 Compactification of Symmetric Spaces [document électronique] / Yves Guivarc'h ; Lizhen Ji ; SpringerLink (Online service) ; John Christopher Taylor (1936-....) . - Bâle (CHE) ; Boston, MA : Birkhäuser, 1998 . - XIV, 284 p : online resource. - (Progress in Mathematics, ISSN 0743-1643; 156) .
ISBN : 978-1-4612-2452-5
Langues : Anglais (eng)
Tags : Mathematics Geometry Topology Résumé : The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view. Key features: * definition and detailed analysis of the Martin compactifications * new geometric Compactification, defined in terms of the Tits building, that coincides with the Martin Compactification at the bottom of the positive spectrum. * geometric, non-inductive, description of the Karpelevic Compactification * study of the well-know isomorphism between the Satake compactifications and the Furstenberg compactifications * systematic and clear progression of topics from geometry to analysis, and finally to random walks The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students Note de contenu : I. Introduction -- Statement of the main new results -- Characterizations of the compactification $${\bar X^{SF}}$$ -- The Karpelevi? compactification $${\bar X^K}$$ -- Fibers of maps between the compactifications -- Application to Brownian motion -- Eigenfunctions and Martin?s method -- Methods of proof -- Open problems -- Conventions -- Study guide -- II. Subalgebras and parabolic subgroups -- The Iwasawa and Cartan decompositions -- Parabolic subgroups -- Subsets of ? and Lie subalgebras -- The Langlands decomposition of PI and the symmetric space XI -- Bruhat decompositions -- III. Geometrical constructions of compactifications -- The conic compactification $${\bar X^c}$$ -- The conical decomposition of a and the Weyl group -- Parabolic subgroups and stabilizers of the points in X(?) -- Flats through the base point and Proposition 3.8 -- The Tits building ?(G) of G and its geometrical realization ?(X) -- The polyhedral compactification of a flat -- The dual cell complex ?*(X) -- The dual cell compactification X ? ?*(X) -- IV. The Satake?Furstenberg compactifications -- Finite dimensional representations -- Weights and highest weights -- Representation and parabolic subgroups -- Satake compactifications -- Furstenberg compactifications -- V. The Karpelevi? compactification -- The Karpelevi? compactification -- Convergence in the Karpelevi? topology restricted to a flat -- The Karpelevi? compactification of a -- The Karpelevi? topology is compact -- The relation between the Karpelevi? compactification, conical and dual cell compactifications -- A characterization of the Karpelevi? compactification -- VI. Martin compactifications -- The Martin compactification -- Convergence of Brownian motion -- Extension of the group action to the Martin compactification -- The Martin compactification for a random walk -- VII. The Martin compactification X ? ?X(?0) -- The Laplacian in horocyclic coordinates -- Generalized horocyclic coordinates and the Laplacian -- Computation of the limit functions: reduction -- The limit of a CI-canonical sequence -- Classification of limit functions and the topology of X ? ?X(?0) -- VIII. The Martin compactification X ? ?X(?) -- The case of X = SL(n, ?)/SU(n) for ? < ?0 -- Computation of the limit functions for a general semisimple group -- Determination of the Martin compactification -- Bounded harmonic functions on X -- An application to convergence of Brownian motion -- IX. An intrinsic approach to the boundaries of X -- The space of closed subgroups -- Limit groups -- Limits of group spheres -- Parabolic subgroups and boundary theory -- The maximal Furstenberg compactification -- X. Compactification via the ground state -- The twisted action -- Compactification of X via the ground state -- XI. Harnack inequality, Martin?s method and the positive spectrum for random walks -- Basic notations -- Cones with compact bases and the Harnack inequality -- Martin?s method for a random walk -- The positive spectrum of a random walk -- The fixed line property -- Formulas for r(p),r0(p) -- Outline of the following chapters -- XII. The Furstenberg boundary and bounded harmonic functions -- Basic notations -- The mean-value property -- Harmonic functions and the mean-value property -- Convergence theorems for harmonic functions -- The Poisson formula for random walks -- XIII. Integral representation of positive eigenfunctions of convolution operators -- The main result of this chapter -- An extension of the main result -- Analytic determination of the minimal eigenfunctions of the Laplacian -- The Busemann cocycle and a geometrical determination of the minimal eigenfunctions of the Laplacian -- Minimal eigenfunctions for random walks -- XIV. Random walks and ground state properties -- Basic definitions and properties -- Convolution -- Spherical functions and minimal eigenfunctions -- Ground state properties -- Random walks, eigenfunctions of the Laplacian and X ? ?X(?0) -- The Martin compactification of X determined by a random walk -- An application to parabolic subgroups -- XV. Extension to semisimple algebraic groups defined over a local field -- Some notations and fundamental properties -- Extension of the main results of Chapters XII, XIII, XIV -- Appendix A -- Appendix B -- List of symbols Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=118670