Book ChapterDOI
Convergence of probability measures
Richard F. Bass
- pp 237-243
Reads0
Chats0
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 anread more
Citations
More filters
Journal ArticleDOI
Asymptotic Optimality of Balanced Routing
Hong Chen,Heng-Qing Ye +1 more
TL;DR: The proposed balanced routing policy for any fixed c ≥ 2 is asymptotically optimal in the sense that it minimizes the workload over all time in the diffusion limit.
Posted Content
Brown-von Neumann-Nash Dynamics: The Continuous Strategy Case
TL;DR: In this paper, it was shown that for continuous payoff functions, the set of rest points of the dynamics coincides with the Nash equilibria of the underlying game, and sufficient conditions for local and global asymptotic stability were identified using concepts developed in evolutionary game theory.
Journal ArticleDOI
A New Evolutionary Algorithm for a Class of Nonlinear Bilevel Programming Problems and Its Global Convergence
TL;DR: A new effective evolutionary algorithm is proposed based on the leader's objective function that has the ability of local search and a new fitness function is proposed that can be easily used to evaluate the quality of different types of potential solutions.
Journal Article
Smooth neighborhood recommender systems
TL;DR: This article proposes a smooth neighborhood recommender in the framework of the latent factor models, and utilizes a “divide-and-conquer” version of the alternating least squares algorithm to achieve scalable computation, and establishes asymptotic results for the proposed method, demonstrating that it achieves superior prediction accuracy.
Journal ArticleDOI
The tail empirical process for long memory stochastic volatility sequences
Rafał Kulik,Philippe Soulier +1 more
TL;DR: In this article, the limiting behaviour of tail empirical processes associated with long memory stochastic volatility models was studied and it was shown that such a process has dichotomous behaviour, according to an interplay between the Hurst parameter and the tail index.
References
More filters
Book
Large Networks and Graph Limits
TL;DR: Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks.
Book
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.
Posted Content
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.
Book ChapterDOI
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.
Journal ArticleDOI
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.