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
Monitoring procedure for parameter change in causal time series
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Geodesics in Brownian surfaces (Brownian maps)
TL;DR: In this article, the scaling limits of random maps on general orientable surfaces with a boundary are defined and the geodesics from a uniformly chosen random point are studied. But the authors use a different approach based on two ingredients: first study typical geodesic and then all geodesici by an "entrapment" strategy.
Journal ArticleDOI
Asymptotics for $p$-value based threshold estimation in regression settings
TL;DR: In this article, the authors investigate the large sample behavior of a $p$-value-based procedure for estimating the threshold level at which a regression function takes off from its baseline value, a problem that frequently arises in environmental statistics, engineering and other related fields.
A central limit theorem for triangular arrays of weakly dependent random variables
TL;DR: In this paper, the authors derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi (1999).
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