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Book ChapterDOI

Convergence of probability measures

Richard F. Bass
- pp 237-243
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TLDR
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 an

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References
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Limit Order Books

TL;DR: This book discusses several models of limit order books and begins by discussing the data to assess their empirical properties, and then moves on to mathematical models in order to reproduce the observed properties.
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Mod-Gaussian convergence and the value distribution of $\zeta(1/2+it)$ and related quantities

TL;DR: In this article, lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian with increasing covariance were derived for random matrices in compact classical groups, as well as for certain families of L-functions over finite fields.
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More uses of exchangeability: representations of complex random structures

David Aldous
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TL;DR: The authors review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures, including processes of stochastic coalescence and fragmentation, continuum random trees, second-order limits of distances in random graphs, isometry classes of metric spaces with probability measures, and dense random graphs.
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On the Grenander Estimator at Zero

TL;DR: In this paper, limit theory for the Grenander estimator of a monotone density near zero is established for the situation when the true density f 0 is unbounded at zero, with different rates of growth to infinity.
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Random Time-Dependent Quantum Walks

TL;DR: In this paper, the authors considered the discrete time unitary dynamics given by a quantum walk on the lattice performed by a particle with internal degree of freedom, called coin state, according to the following iterated rule.