An Roussas, George G. creator author. text bibliography Electronic books. ne 2014 2nd ed. monographic eng

electronic resource

1 online resource "In this introductory chapter, the concepts of a field and of a [sigma]-field are introduced, they are illustrated bymeans of examples, and some relevant basic results are derived. Also, the concept of a monotone class is defined and its relationship to certain fields and [sigma]-fields is investigated. Given a collection of measurable spaces, their product space is defined, and some basic properties are established. The concept of a measurable mapping is introduced, and its relation to certain [sigma]-fields is studied. Finally, it is shown that any random variable is the pointwise limit of a sequence of simple random variables"-- Certain classes of sets, measurability, and pointwise approximation -- Definition and construction of a measure and its basic properties -- Some modes of convergence of sequences of random variables and their relationships -- The integral of a random variable and its basic properties -- Standard convergence theorems, the Fubini theorem -- Standard moment and probability inequalities, convergence in the rth mean and its implications -- The Hahn-Jordan decomposition theorem, the Lebesgue decomposition theorem, and the Radon-Nikodym theorem -- Distribution functions and their basic properties, Helly-Bray type results -- Conditional expectation and conditional probability, and related properties and results -- Independence -- Topics from the theory of characteristic functions -- The central limit problem: the centered case -- The central limit problem: the noncentered case -- Topics from sequences of independent random variables -- Topics from Ergodic theory -- Two cases of statistical inference: estimation of a real-valued parameter, nonparametric estimation of a probability density function -- Appendixes: A. Brief review of chapters 1-16 -- B. Brief review of Riemann-Stieltjes integral -- C. Notation and abbreviations. by George G. Roussas. Includes bibliographical references and index. Probabilities Measure theory Measure theory Probabilities QA273 .R864 2014eb 519.2 Roussas, George G. Second edition (DLC) 2014007243 (OCoLC)868642456 9780128002902 0128002905 http://www.sciencedirect.com/science/book/9780128000427 http://www.sciencedirect.com/science/book/9780128000427 OPELS 140414 20190328114807.0 ocn876589188 eng