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
Citations
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References
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Multidimensional Stochastic Processes as Rough Paths
Peter K. Friz,Nicolas B. Victoir +1 more
TL;DR: Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations as mentioned in this paper, and it has been used extensively in the analysis of partial differential equations.
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Certifying Some Distributional Robustness with Principled Adversarial Training
TL;DR: In this paper, a training procedure that augments model parameter updates with worst-case perturbations of training data is proposed to guarantee moderate levels of robustness with little computational or statistical cost relative to empirical risk minimization.
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Pure exploration in multi-armed bandits problems
TL;DR: The main result is that the required exploration-exploitation trade-offs are qualitatively different, in view of a general lower bound on the simple regret in terms of the cumulative regret.
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On the limits of communication with low-precision analog-to-digital conversion at the receiver
TL;DR: This work evaluates the communication limits imposed by low-precision ADC for transmission over the real discrete-time additive white Gaussian noise (AWGN) channel, under an average power constraint on the input.