Book ChapterDOI
Convergence of probability measures
Richard F. Bass
- pp 237-243
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Weakconvergence methods in metric spaces were studied in this article, with applications sufficient to show their power and utility, and the results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables.Abstract:
The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is anread more
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References
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Journal ArticleDOI
Central limit theorem for a Stratonovich integral with Malliavin calculus
Daniel Harnett,David Nualart +1 more
TL;DR: In this article, the authors present the publisher's version, also available electronically from http://projecteuclid.aop/1372859768, also available on the project's website.
Journal ArticleDOI
Variational convergence: Approximation and existence of equilibria in discontinuous games
TL;DR: The notion of convergence is used to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.
Journal ArticleDOI
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TL;DR: In this article, the authors studied the asymptotic study of adaptive group sequential designs in the case of randomized clinical trials (RCTs) with binary treatment, binary outcome and no covariate.
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Central limit theorem for the multilevel Monte Carlo Euler method
Mohamed Ben Alaya,Ahmed Kebaier +1 more
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Prediction of tree-size distributions and inventory variables from cumulants of canopy height distributions
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