[PDF] Behavior Of Convolution Sequences Of A Family Of Probability Measures On The Interval 0 Infinity eBook

Author : A. Mukherhea
Publisher :
Page : 14 pages
File Size : 36,92 MB
Release : 1973
Category :
ISBN :

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In this paper, the authors, consider a result due to M. Rosenblatt which is frequently useful in the theory of random walks. His result states that if mu is a regular probability measure on a compact semigroup S which is generated by the support of mu, then given any open set O containing an ideal of S, (mu sup n)(O) converges to 1 as n nears infinity. The essential contribution of this paper is an example of an interesting family of probability measures on the interval(0, infinity) which shows that Rosenblatt's theorem cannot be extended to a general locally compact semigroup. Of further significance in this paper is the indicated relationship between the Central Limit Theorem of probability theory on the one hand and polynomial approximation of the exponential function on the other.

Author : Terence Tao
Publisher : American Mathematical Soc.
Page : 206 pages
File Size : 14,76 MB
Release : 2021-09-03
Category : Education
ISBN : 1470466406

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This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

Author : Patrick Billingsley
Publisher : John Wiley & Sons
Page : 612 pages
File Size : 16,92 MB
Release : 2017
Category :
ISBN : 9788126517718

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Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory.· Probability· Measure· Integration· Random Variables and Expected Values· Convergence of Distributions· Derivatives and Conditional Probability· Stochastic Processes

Author : Luigi Ambrosio
Publisher : Springer Science & Business Media
Page : 333 pages
File Size : 45,6 MB
Release : 2008-10-29
Category : Mathematics
ISBN : 376438722X

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The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.