Limit Theorems of Probability Theory
Limit Theorems of Probability Theory

Author: Yu.V. Prokhorov

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 280

ISBN-13: 3662041723

DOWNLOAD EBOOK

A collection of research level surveys on certain topics in probability theory by a well-known group of researchers. The book will be of interest to graduate students and researchers.

Limit Theorems for Multi-Indexed Sums of Random Variables
Limit Theorems for Multi-Indexed Sums of Random Variables

Author: Oleg Klesov

Publisher: Springer

Published: 2014-10-13

Total Pages: 495

ISBN-13: 3662443880

DOWNLOAD EBOOK

Presenting the first unified treatment of limit theorems for multiple sums of independent random variables, this volume fills an important gap in the field. Several new results are introduced, even in the classical setting, as well as some new approaches that are simpler than those already established in the literature. In particular, new proofs of the strong law of large numbers and the Hajek-Renyi inequality are detailed. Applications of the described theory include Gibbs fields, spin glasses, polymer models, image analysis and random shapes. Limit theorems form the backbone of probability theory and statistical theory alike. The theory of multiple sums of random variables is a direct generalization of the classical study of limit theorems, whose importance and wide application in science is unquestionable. However, to date, the subject of multiple sums has only been treated in journals. The results described in this book will be of interest to advanced undergraduates, graduate students and researchers who work on limit theorems in probability theory, the statistical analysis of random fields, as well as in the field of random sets or stochastic geometry. The central topic is also important for statistical theory, developing statistical inferences for random fields, and also has applications to the sciences, including physics and chemistry.

Limit Distributions for Sums of Independent Random Variables
Limit Distributions for Sums of Independent Random Variables

Author: B. V. Gnedenko

Publisher:

Published: 2021-08-05

Total Pages: 276

ISBN-13: 9781684225798

DOWNLOAD EBOOK

2021 Reprint of the 1954 Edition. Facsimile of the original edition and not reproduced with Optical Recognition. This treatise on the fundamental limit theorems in probability theory is strong on mathematical rigor, but the presentation is equally distinguished by clarity and elegance. With broad perspectives on the development from the law of large numbers (Bernoulli, 1713) and the limit theorems of de Moivre (1730), Laplace (1812), and Poisson (1837), over the important progress made by Chebyshev (1867, 1890), Lyapunov (1901), and Lindeberg (1922), the book focuses on the progress experts had made on the subject up to the time of publication. This book may be considered a genuine classic. Of general interest is the material in the two first chapters, which may serve as a basis for any rigorous course in probability theory. The axiomatic foundation of the theory given by Kolmogorov in 1933 is here somewhat modified, the probability distributions being specified so as to form a perfect measure, a restriction which removes certain intricacies, notably in the treatment of conditional probabilities and expectations. The careful translation is based on the Russian original (1949. The two appendixes and some fifty extra footnotes give clarifying and instructive remarks.

Random Summation
Random Summation

Author: Boris V. Gnedenko

Publisher: CRC Press

Published: 2020-07-24

Total Pages: 282

ISBN-13: 1000141179

DOWNLOAD EBOOK

This book provides an introduction to the asymptotic theory of random summation, combining a strict exposition of the foundations of this theory and recent results. It also includes a description of its applications to solving practical problems in hardware and software reliability, insurance, finance, and more. The authors show how practice interacts with theory, and how new mathematical formulations of problems appear and develop. Attention is mainly focused on transfer theorems, description of the classes of limit laws, and criteria for convergence of distributions of sums for a random number of random variables. Theoretical background is given for the choice of approximations for the distribution of stock prices or surplus processes. General mathematical theory of reliability growth of modified systems, including software, is presented. Special sections deal with doubling with repair, rarefaction of renewal processes, limit theorems for supercritical Galton-Watson processes, information properties of probability distributions, and asymptotic behavior of doubly stochastic Poisson processes. Random Summation: Limit Theorems and Applications will be of use to specialists and students in probability theory, mathematical statistics, and stochastic processes, as well as to financial mathematicians, actuaries, and to engineers desiring to improve probability models for solving practical problems and for finding new approaches to the construction of mathematical models.

Limit Theorems For Associated Random Fields And Related Systems
Limit Theorems For Associated Random Fields And Related Systems

Author: Alexander Bulinski

Publisher: World Scientific

Published: 2007-09-05

Total Pages: 447

ISBN-13: 9814474576

DOWNLOAD EBOOK

This volume is devoted to the study of asymptotic properties of wide classes of stochastic systems arising in mathematical statistics, percolation theory, statistical physics and reliability theory. Attention is paid not only to positive and negative associations introduced in the pioneering papers by Harris, Lehmann, Esary, Proschan, Walkup, Fortuin, Kasteleyn and Ginibre, but also to new and more general dependence conditions. Naturally, this scope comprises families of independent real-valued random variables. A variety of important results and examples of Markov processes, random measures, stable distributions, Ising ferromagnets, interacting particle systems, stochastic differential equations, random graphs and other models are provided. For such random systems, it is worthwhile to establish principal limit theorems of the modern probability theory (central limit theorem for random fields, weak and strong invariance principles, functional law of the iterated logarithm etc.) and discuss their applications.There are 434 items in the bibliography.The book is self-contained, provides detailed proofs, for reader's convenience some auxiliary results are included in the Appendix (e.g. the classical Hoeffding lemma, basic electric current theory etc.).