## Probability Theory and Mathematical Statistics. Vol. 1

**Author: ** B. Grigelionis , Yu. V. Prohorov , V. V. Sazonov , V. Statulevičius

**Number of pages:**652

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Presents an introduction to probability and statistics. This book covers topics that include the axiomatic setup of probability theory, polynomial distribution, finite Markov chains, distribution functions and convolution, the laws of large numbers (weak and strong), characteristic functions, the central limit theorem, and Markov processes.

This book presents a rigorous exposition of probability theory for a variety of applications. The first part of the book is a self-contained account of the fundamentals. Material suitable for advanced study is then developed from the basic concepts. Emphasis is placed on examples, sound interpretation of results and scope for applications.A distinctive feature of the book is that it discusses modern applications seldom covered in traditional texts. Two cases in point are risk theory (or comparison of distributions) and stochastic optimization. The book also includes some recent developments of probability theory, for example limit theorems for sums of dependent variables, nonlinear and nonclassical limit theorems. Simplified proofs and a unified approach to the exposition of many results are other key features.The book may be used as a textbook for graduate students and advanced undergraduates, and as a work of reference.

This book is an excellent introduction to probability theory for students who have some general experience from university-level mathematics, in particular, analysis. It would be suitable for reading in conjunction with a second or third year course in probability theory. Besides the standard material, the author has included sections on special topics, for example percolation and statistical mechanics, which are direct applications of the theory.

The book is an introduction to modern probability theory written by one of the famous experts in this area. Readers will learn about the basic concepts of probability and its applications, preparing them for more advanced and specialized works.

The topics treated fall into three main groups, all of which deal with classical problems which originated in the work of Kolmogorov. The first section looks at probability limit theorems, the second deals with stochastic analysis, and the final part presents some papers on non-parametric and semi-parametric models of mathematical statistics and asymptotic problems. The contributions come from some of the foremost mathematicians in the world today, making for a truly international collection of papers, permeated with the influence of Kolmogorov's works.

The Probability Theory of Patterns and Runs has had a long and distinguished history, starting with the work of de Moivre in the 18th century and that of von Mises in the early 1920's, and continuing with the renewal-theoretic results in Feller's classic text An Introduction to Probability Theory and its Applications, Volume 1. It is worthwhile to note, in particular, that de Moivre, in the third edition of The Doctrine of Chances (1756, reprinted by Chelsea in 1967, pp. 254-259), provides the generating function for the waiting time for the appearance of k consecutive successes. During the 1940's, statisticians such as Mood, Wolfowitz, David and Mosteller studied the distribution theory, both exact and asymptotic, of run-related statistics, thereby laying the foundation for several exact run tests. In the last two decades or so, the theory has seen an impressive re-emergence, primarily due to important developments in Molecular Biology, but also due to related research thrusts in Reliability Theory, Distribution Theory, Combinatorics, and Statistics.

"This book is a concise introduction to modern probability theory an certain of its ramifications. By deliberate succinctness of style and judicious selection of topics, it manages to be both fast-moving and self-contained." -- Editor's preface.

The concept of probability. Sequences of independent trials. Markov chains. Random variables and distribution functions; Numerical characteristics of random variables. The law of large numbers. Characteristics functions. The classical limit theorem. The theory of infinitely divisible distribution. The theory of stochastic processes. Elements of queueing theory.

This book provides a systematic, self-sufficient and yet short presentation of the mainstream topics on introductory Probability Theory with some selected topics from Mathematical Statistics. It is suitable for a 10- to 14-week course for second- or third-year undergraduate students in Science, Mathematics, Statistics, Finance, or Economics, who have completed some introductory course in Calculus. There is a sufficient number of problems and solutions to cover weekly tutorials.

This two-volume textbook set presents a systematic treatment of probability from the ground up. It begins with intuitive ideas and gradually develops more sophisticated subjects, such as random walks, martingales, Markov chains, the measure-theoretic foundations of probability theory, weak convergence of probability measures, and the central limit theorem. Numerous detailed examples and exercises are provided throughout. In addition to the thorough presentation of probability, a historical exposition is presented that documents how the mathematical theory of probability has developed. The two volumes that compose this set together constitute the third English edition of the author’s classic, Probability. The first volume is particularly suitable for a graduate course on probability aimed at mathematics, statistics, or engineering students, and the second volume would be ideal for a graduate course on random process.

The aim of this book is to serve as a reference text to provide an orientation in the enormous material which probability theory has accumulated so far. The book mainly treats such topics like the founda tions of probability theory, limit theorems and random processes. The bibliography gives a list of the main textbooks on probability theory and its applications. By way of exception some references are planted into the text to recent papers which in our opinion did not find in monographs the attention they deserved (in this connection we do not at all want to attribute any priority to one or the other author). Some references indicate the immediate use of the material taken from the paper in question. In the following we recommend some selected literature, together with indications of the corresponding sections of the present reference book. The textbook by B. V. Gnedenko, "Lehrbuch der Wahrscheinlichkeits theorie " , Akademie-Verlag, Berlin 1957, and the book by W. Feller, "IntroductioI). to Probability Theory and its Applications", Wiley, 2. ed., New York 1960 (Chapter I, § 1 of Chapter V) may serve as a first introduction to the various problems of probability theory. A large...

The book is an introduction to modern probability theory written by one of the famous experts in this area. Readers will learn about the basic concepts of probability and its applications, preparing them for more advanced and specialized works.

Basic concepts; Random variables; Expectation; Conditional probability and expectation; Characteristic functions; Infinite sequences of Random variables; Markov chains; Introduction to statistics.

This clear exposition begins with basic concepts and moves on to combination of events, dependent events and random variables, Bernoulli trials and the De Moivre-Laplace theorem, and more. Includes 150 problems, many with answers.

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