Applications of Characteristic Functions
Author | : Eugene Lukacs |
Publisher | : |
Total Pages | : 214 |
Release | : 1964 |
Genre | : Characteristic functions |
ISBN | : |
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Author | : Eugene Lukacs |
Publisher | : |
Total Pages | : 214 |
Release | : 1964 |
Genre | : Characteristic functions |
ISBN | : |
Author | : Zoltán Sasvári |
Publisher | : Walter de Gruyter |
Total Pages | : 376 |
Release | : 2013-03-22 |
Genre | : Mathematics |
ISBN | : 3110223996 |
In a certain sense characteristic functions and correlation functions are the same, the common underlying concept is positive definiteness. Many results in probability theory, mathematical statistics and stochastic processes can be derived by using these functions. While there are books on characteristic functions of one variable, books devoting some sections to the multivariate case, and books treating the general case of locally compact groups, interestingly there is no book devoted entirely to the multidimensional case which is extremely important for applications. This book is intended to fill this gap at least partially. It makes the basic concepts and results on multivariate characteristic and correlation functions easily accessible to both students and researchers in a comprehensive manner. The first chapter presents basic results and should be read carefully since it is essential for the understanding of the subsequent chapters. The second chapter is devoted to correlation functions, their applications to stationary processes and some connections to harmonic analysis. In Chapter 3 we deal with several special properties, Chapter 4 is devoted to the extension problem while Chapter 5 contains a few applications. A relatively large appendix comprises topics like infinite products, functional equations, special functions or compact operators.
Author | : Emil Joseph Polak |
Publisher | : |
Total Pages | : 134 |
Release | : 1951 |
Genre | : Probabilities |
ISBN | : |
Author | : Solomon Kullback |
Publisher | : |
Total Pages | : 202 |
Release | : 1940* |
Genre | : Characteristic functions |
ISBN | : |
Author | : Marco Taboga |
Publisher | : Createspace Independent Publishing Platform |
Total Pages | : 670 |
Release | : 2017-12-08 |
Genre | : Mathematical statistics |
ISBN | : 9781981369195 |
The book is a collection of 80 short and self-contained lectures covering most of the topics that are usually taught in intermediate courses in probability theory and mathematical statistics. There are hundreds of examples, solved exercises and detailed derivations of important results. The step-by-step approach makes the book easy to understand and ideal for self-study. One of the main aims of the book is to be a time saver: it contains several results and proofs, especially on probability distributions, that are hard to find in standard references and are scattered here and there in more specialistic books. The topics covered by the book are as follows. PART 1 - MATHEMATICAL TOOLS: set theory, permutations, combinations, partitions, sequences and limits, review of differentiation and integration rules, the Gamma and Beta functions. PART 2 - FUNDAMENTALS OF PROBABILITY: events, probability, independence, conditional probability, Bayes' rule, random variables and random vectors, expected value, variance, covariance, correlation, covariance matrix, conditional distributions and conditional expectation, independent variables, indicator functions. PART 3 - ADDITIONAL TOPICS IN PROBABILITY THEORY: probabilistic inequalities, construction of probability distributions, transformations of probability distributions, moments and cross-moments, moment generating functions, characteristic functions. PART 4 - PROBABILITY DISTRIBUTIONS: Bernoulli, binomial, Poisson, uniform, exponential, normal, Chi-square, Gamma, Student's t, F, multinomial, multivariate normal, multivariate Student's t, Wishart. PART 5 - MORE DETAILS ABOUT THE NORMAL DISTRIBUTION: linear combinations, quadratic forms, partitions. PART 6 - ASYMPTOTIC THEORY: sequences of random vectors and random variables, pointwise convergence, almost sure convergence, convergence in probability, mean-square convergence, convergence in distribution, relations between modes of convergence, Laws of Large Numbers, Central Limit Theorems, Continuous Mapping Theorem, Slutsky's Theorem. PART 7 - FUNDAMENTALS OF STATISTICS: statistical inference, point estimation, set estimation, hypothesis testing, statistical inferences about the mean, statistical inferences about the variance.
Author | : Wlodzimierz Bryc |
Publisher | : Springer Science & Business Media |
Total Pages | : 142 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461225604 |
This book is a concise presentation of the normal distribution on the real line and its counterparts on more abstract spaces, which we shall call the Gaussian distributions. The material is selected towards presenting characteristic properties, or characterizations, of the normal distribution. There are many such properties and there are numerous rel evant works in the literature. In this book special attention is given to characterizations generated by the so called Maxwell's Theorem of statistical mechanics, which is stated in the introduction as Theorem 0.0.1. These characterizations are of interest both intrin sically, and as techniques that are worth being aware of. The book may also serve as a good introduction to diverse analytic methods of probability theory. We use characteristic functions, tail estimates, and occasionally dive into complex analysis. In the book we also show how the characteristic properties can be used to prove important results about the Gaussian processes and the abstract Gaussian vectors. For instance, in Section 5.4 we present Fernique's beautiful proofs of the zero-one law and of the integrability of abstract Gaussian vectors. The central limit theorem is obtained via characterizations in Section 7.3.
Author | : Gordon Lynn Harrison |
Publisher | : |
Total Pages | : 74 |
Release | : 1968 |
Genre | : Characteristic functions |
ISBN | : |
Author | : Zhaoping Jiang |
Publisher | : |
Total Pages | : 170 |
Release | : 1993 |
Genre | : Characteristic functions |
ISBN | : |