Central limit theorem

About: Central limit theorem is a research topic. Over the lifetime, 10119 publications have been published within this topic receiving 253995 citations. The topic is also known as: CLT.

Convergence of stochastic processes

Abstract: I Functional on Stochastic Processes.- 1. Stochastic Processes as Random Functions.- Notes.- Problems.- II Uniform Convergence of Empirical Measures.- 1. Uniformity and Consistency.- 2. Direct Approximation.- 3. The Combinatorial Method.- 4. Classes of Sets with Polynomial Discrimination.- 5. Classes of Functions.- 6. Rates of Convergence.- Notes.- Problems.- III Convergence in Distribution in Euclidean Spaces.- 1. The Definition.- 2. The Continuous Mapping Theorem.- 3. Expectations of Smooth Functions.- 4. The Central Limit Theorem.- 5. Characteristic Functions.- 6. Quantile Transformations and Almost Sure Representations.- Notes.- Problems.- IV Convergence in Distribution in Metric Spaces.- 1. Measurability.- 2. The Continuous Mapping Theorem.- 3. Representation by Almost Surely Convergent Sequences.- 4. Coupling.- 5. Weakly Convergent Subsequences.- Notes.- Problems.- V The Uniform Metric on Spaces of Cadlag Functions.- 1. Approximation of Stochastic Processes.- 2. Empirical Processes.- 3. Existence of Brownian Bridge and Brownian Motion.- 4. Processes with Independent Increments.- 5. Infinite Time Scales.- 6. Functional of Brownian Motion and Brownian Bridge.- Notes.- Problems.- VI The Skorohod Metric on D(0, ?).- 1. Properties of the Metric.- 2. Convergence in Distribution.- Notes.- Problems.- VII Central Limit Theorems.- 1. Stochastic Equicontinuity.- 2. Chaining.- 3. Gaussian Processes.- 4. Random Covering Numbers.- 5. Empirical Central Limit Theorems.- 6. Restricted Chaining.- Notes.- Problems.- VIII Martingales.- 1. A Central Limit Theorem for Martingale-Difference Arrays.- 2. Continuous Time Martingales.- 3. Estimation from Censored Data.- Notes.- Problems.- Appendix A Stochastic-Order Symbols.- Appendix B Exponential Inequalities.- Notes.- Problems.- Appendix C Measurability.- Notes.- Problems.- References.- Author Index.

Sums of Independent Random Variables

Abstract: I. Probability Distributions and Characteristic Functions.- 1. Random variables and probability distributions.- 2. Characteristic functions.- 3. Inversion formulae.- 4. The convergence of sequences of distributions and characteristic functions.- 5. Supplement.- II. Infinitely Divisible Distributions.- 1. Definition and elementary properties of infinitely divisible distributions.- 2. Canonical representation of infinitely divisible characteristic functions.- 3. An auxiliary theorem.- 4. Supplement.- III. Some Inequalities for the Distribution of Sums of Independent Random Variables.- 1. Concentration functions.- 2. Inequalities for the concentration functions of sums of independent random variables.- 3. Inequalities for the distribution of the maximum of sums of independent random variables.- 4. Exponential estimates for the distributions of sums of independent random variables.- 5. Supplement.- IV. Theorems on Convergence to Infinitely Divisible Distributions.- 1. Infinitely divisible distributions as limits of the distributions of sums of independent random variables.- 2. Conditions for convergence to a given infinitely divisible distribution.- 3. Limit distributions of class L and stable distributions.- 4. The central limit theorem.- 5. Supplement.- V. Estimates of the Distance Between the Distribution of a Sum of Independent Random Variables and the Normal Distribution.- 1. Estimating the nearness of functions of bounded variation by the nearness of their Fourier-Stieltjes transforms.- 2. The Esseen and Berry-Esseen inequalities.- 3. Generalizations of Esseen's inequality.- 4. Non-uniform estimates.- 5. Supplement.- VI. Asymptotic Expansions in the Central Limit Theorem.- 1. Formal construction of the expansions.- 2 Auxiliary propositions.- 3. Asymptotic expansions of the distribution function of a sum of independent identically distributed random variables.- 4. Asymptotic expansions of the distribution function of a sum of independent non-identically distributed random variables, and of the derivatives of this function.- 5. Supplement.- VII. Local Limit Theorems.- 1. Local limit theorems for lattice distributions.- 2. Local limit theorems for densities.- 3. Asymptotic expansions in local limit theorems.- 4. Supplement.- VIII. Probabilities of Large Deviations.- 1. Introduction.- 2. Asymptotic relations connected with Cramer's series.- 3. Necessary and sufficient conditions for normal convergence in power zones.- 4. Supplement.- IX. Laws of Large Numbers.- 1. The weak law of large numbers.- 2. Convergence of series of independent random variables.- 3. The strong law of large numbers.- 4. Convergence rates in the laws of large numbers.- 5. Supplement.- X. The Law of the Iterated Logarithm.- 1. Kolmogorov's theorem.- 2. Generalization of Kolmogorov's theorem.- 3. The central limit theorem and the law of the iterated logarithm.- 4. Supplement.- Notes on Sources in the Literature.- References.- Subject Indes.- Table of Symbols and Abbreviations.

Probability in Banach Spaces: Isoperimetry and Processes

Abstract: Notation.- 0. Isoperimetric Background and Generalities.- 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- 2. Generalities on Banach Space Valued Random Variables and Random Processes.- I. Banach Space Valued Random Variables and Their Strong Limiting Properties.- 3. Gaussian Random Variables.- 4. Rademacher Averages.- 5. Stable Random Variables.- 6 Sums of Independent Random Variables.- 7. The Strong Law of Large Numbers.- 8. The Law of the Iterated Logarithm.- II. Tightness of Vector Valued Random Variables and Regularity of Random Processes.- 9. Type and Cotype of Banach Spaces.- 10. The Central Limit Theorem.- 11. Regularity of Random Processes.- 12. Regularity of Gaussian and Stable Processes.- 13. Stationary Processes and Random Fourier Series.- 14. Empirical Process Methods in Probability in Banach Spaces.- 15. Applications to Banach Space Theory.- References.

Asymptotic theory for econometricians

Abstract: The Linear Model and Instrumental Variables Estimators. Consistency. Laws of Large Numbers. Asymptotic Normality. Central Limit Theory. Estimating Asymptotic Covariance Matrices. Functional Central Limit Theory and Applications. Directions for Further Study. Solution Set. References. Index.

Entropy, large deviations, and statistical mechanics

Abstract: I: Large Deviations and Statistical Mechanics.- I. Introduction to Large Deviations.- I.1. Overview.- I.2. Large Deviations for I.I.D. Random Variables with a Finite State Space.- I.3. Levels-1 and 2 for Coin Tossing.- I.4. Levels-1 and 2 for I.I.D. Random Variables with a Finite State Space.- I.S. Level-3: Empirical Pair Measure.- I.6. Level-3: Empirical Process.- I.7. Notes.- I.B. Problems.- II. Large Deviation Property and Asymptotics of Integrals.- II.1. Introduction.- II.2. Levels-1, 2, and 3 Large Deviations for I.I.D. Random Vectors.- II.3. The Definition of Large Deviation Property.- II.4. Statement of Large Deviation Properties for Levels-1, 2, and 3.- II.5. Contraction Principles.- II.6. Large Deviation Property for Random Vectors and Exponential Convergence.- II.7. Varadhan's Theorem on the Asymptotics of Integrals.- II.8. Notes.- II.9. Problems.- III. Large Deviations and the Discrete Ideal Gas.- III.1. Introduction.- III.2. Physics Prelude: Thermodynamics.- III.3. The Discrete Ideal Gas and the Microcanonical Ensemble.- III.4. Thermodynamic Limit, Exponential Convergence, and Equilibrium Values.- III.5. The Maxwell-Boltzmann Distribution and Temperature.- III.6. The Canonical Ensemble and Its Equivalence with the Microcanonical Ensemble.- III.7. A Derivation of a Thermodynamic Equation.- III.8. The Gibbs Variational Formula and Principle.- III.9. Notes.- III.10. Problems.- IV. Ferromagnetic Models on ?.- IV.1. Introduction.- IV.2. An Overview of Ferromagnetic Models.- IV.3. Finite-Volume Gibbs States on ?.- IV.4. Spontaneous Magnetization for the Curie-Weiss Model.- IV.5. Spontaneous Magnetization for General Ferromagnets on ?.- IV.6. Infinite-Volume Gibbs States and Phase Transitions.- IV.7. The Gibbs Variational Formula and Principle.- IV.8. Notes.- IV.9. Problems.- V. Magnetic Models on ?D and on the Circle.- V.1. Introduction.- V.2. Finite-Volume Gibbs States on ?D, D ? 1.- V.3. Moment Inequalities.- V.4. Properties of the Magnetization and the Gibbs Free Energy.- V.5. Spontaneous Magnetization on ?D, D ? 2, Via the Peierls Argument.- V.6. Infinite-Volume Gibbs States and Phase Transitions.- V.7. Infinite-Volume Gibbs States and the Central Limit Theorem.- V.8. Critical Phenomena and the Breakdown of the Central Limit Theorem.- V.9. Three Faces of the Curie-Weiss Model.- V.10. The Circle Model and Random Waves.- V.11. A Postscript on Magnetic Models.- V.12. Notes.- V.13. Problems.- II: Convexity and Proofs of Large Deviation Theorems.- VI. Convex Functions and the Legendre-Fenchel Transform.- VI.1. Introduction.- VI.2. Basic Definitions.- VI.3. Properties of Convex Functions.- VI.4. A One-Dimensional Example of the Legendre-Fenchel Transform.- VI.5. The Legendre-Fenchel Transform for Convex Functions on ?d.- VI.6. Notes.- VI.7. Problems.- VII. Large Deviations for Random Vectors.- VII.1. Statement of Results.- VII.2. Properties of IW.- VII.3. Proof of the Large Deviation Bounds for d = 1.- VII.4. Proof of the Large Deviation Bounds for d ? 1.- VII.5. Level-1 Large Deviations for I.I.D. Random Vectors.- VII.6. Exponential Convergence and Proof of Theorem II.6.3.- VII.7. Notes.- VII.8. Problems.- VIII. Level-2 Large Deviations for I.I.D. Random Vectors.- VIII.1. Introduction.- VIII.2. The Level-2 Large Deviation Theorem.- VIII.3. The Contraction Principle Relating Levels-1 and 2 (d = 1).- VIII.4. The Contraction Principle Relating Levels-1 and 2 (d ? 2).- VIII.5. Notes.- VIII.6. Problems.- IX. Level-3 Large Deviations for I.I.D. Random Vectors.- IX.1. Statement of Results.- IX.2. Properties of the Level-3 Entropy Function.- IX.3. Contraction Principles.- IX.4. Proof of the Level-3 Large Deviation Bounds.- IX.5. Notes.- IX.6. Problems.- Appendices.- Appendix A: Probability.- A.1. Introduction.- A.2. Measurability.- A.3. Product Spaces.- A.4. Probability Measures and Expectation.- A.S. Convergence of Random Vectors.- A.6. Conditional Expectation, Conditional Probability, and Regular Conditional Distribution.- A.7. The Kolmogorov Existence Theorem.- A.8. Weak Convergence of Probability Measures on a Metric Space.- Appendix B: Proofs of Two Theorems in Section II.7.- B.1. Proof of Theorem II.7.1.- B.2. Proof of Theorem II.7.2.- Appendix C: Equivalent Notions of Infinite-Volume Measures for Spin Systems.- C.1. Introduction.- C.2. Two-Body Interactions and Infinite-Volume Gibbs States.- C.3. Many-Body Interactions and Infinite-Volume Gibbs States.- C.4. DLR States.- C.5. The Gibbs Variational Formula and Principle.- C.6. Solution of the Gibbs Variational Formula for Finite-Range Interactions on ?.- Appendix D: Existence of the Specific Gibbs Free Energy.- D.1. Existence Along Hypercubes.- D.2. An Extension.- List of Frequently Used Symbols.- References.- Author Index.