Topic
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.
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TL;DR: This paper focuses on the analysis of the Fouriersche Integrate polynomial, which is a very simple and straightforward way to solve the inequality of the following type: For α ≥ 1, β ≥ 1 using LaSalle's inequality.
Abstract: * The research for this paper was supported by the United States Air Force under Contract No. AF18(600-685) monitored by the Office of Scientific Research. 1 A discussion of the problem along with the necessary references will be found in Harry Pollard, "The Harmonic Analysis of Bounded Functions," Duke Math. / . , 20, 499-512, 1953. 2 See ibid. 3 See S. Bochner, Fouriersche Integrate (Leipzig, 1932), p. 33. 4 The Levitan polynomial. See N.I. Achieser, Approximationstheorie (Berlin, 1953), p. 146.
1,226 citations
Book•
05 Apr 1976
TL;DR: In this paper, the Euler-MacLaurin sum formula for functions of several variables has been applied to the problem of convergence of probability measures and uniformity classes, and it has been shown that it is possible to obtain strong convergence for continuous, singular, and discrete probability measures.
Abstract: Preface to the Classics Edition Preface 1. Weak convergence of probability measures and uniformity classes 2. Fourier transforms and expansions of characteristic functions 3. Bounds for errors of normal approximation 4. Asymptotic expansions-nonlattice distributions 5. Asymptotic expansions-lattice distributions 6. Two recent improvements 7. An application of Stein's method Appendix A.1. Random vectors and independence Appendix A.2. Functions of bounded variation and distribution functions Appendix A.3. Absolutely continuous, singular, and discrete probability measures Appendix A.4. The Euler-MacLaurin sum formula for functions of several variables References Index.
1,125 citations
TL;DR: In this article, a nonlinear transformation of two independent uniform random variables into one stable random variable is presented, which is a continuous function of each of the uniform random variable, and of α and a modified skewness parameter β' throughout their respective permissible ranges.
Abstract: A new algorithm is presented for simulating stable random variables on a digital computer for arbitrary characteristic exponent α(0 < α ≤ 2) and skewness parameter β(-1 ≤ β ≤ 1). The algorithm involves a nonlinear transformation of two independent uniform random variables into one stable random variable. This stable random variable is a continuous function of each of the uniform random variables, and of α and a modified skewness parameter β' throughout their respective permissible ranges.
1,124 citations
Book•
11 Sep 2006TL;DR: In this article, the authors present a case study of non-normal distribution and non-commutative joint distributions and define a set of basic combinatorics, such as non-crossing partitions, sum-of-free random variables, and products of free random variables.
Abstract: Part I. Basic Concepts: 1. Non-commutative probability spaces and distributions 2. A case study of non-normal distribution 3. C*-probability spaces 4. Non-commutative joint distributions 5. Definition and basic properties of free independence 6. Free product of *-probability spaces 7. Free product of C*-probability spaces Part II. Cumulants: 8. Motivation: free central limit theorem 9. Basic combinatorics I: non-crossing partitions 10. Basic Combinatorics II: Mobius inversion 11. Free cumulants: definition and basic properties 12. Sums of free random variables 13. More about limit theorems and infinitely divisible distributions 14. Products of free random variables 15. R-diagonal elements Part III. Transforms and Models: 16. The R-transform 17. The operation of boxed convolution 18. More on the 1-dimensional boxed convolution 19. The free commutator 20. R-cyclic matrices 21. The full Fock space model for the R-transform 22. Gaussian Random Matrices 23. Unitary Random Matrices Notes and Comments Bibliography Index.
1,097 citations
Book•
01 Dec 1988TL;DR: Reprint of entire volume Discrete probability distributions ( chapter 1) Continuous probability densities ( chapter 2) Combinatorics ( chapter 3) Conditional probability ( chapter 4) Important distributions and densities( chapter 5) Expected value and variance ( chapter 6) Sums of independent random variables ( chapter 7)
Abstract: Reprint of entire volume Discrete probability distributions (Chapter 1) Continuous probability densities (Chapter 2) Combinatorics (Chapter 3) Conditional probability (Chapter 4) Important distributions and densities (Chapter 5) Expected value and variance (Chapter 6) Sums of independent random variables (Chapter 7) Law of large numbers (Chapter 8) Central limit theorem (Chapter 9) Generating functions (Chapter 10) Markov chains (Chapter 11) Random walks (Chapter 12) Appendices Index.
1,081 citations