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
Sequential testing for the stability of high frequency portfolio betas
TL;DR: In this paper, a modified functional CAPM and sequential monitoring procedures are proposed to test for the constancy of the portfolio betas in a simulation study and an application to S&P 100 data.
Journal ArticleDOI
Theory and inference for a class of nonlinear models with application to time series of counts
Richard A. Davis,Heng Liu +1 more
TL;DR: In this paper, the authors employ an iterated random function approach and a special coupling technique to show that, under suitable conditions on the parameter space, the conditional mean process is a geometric moment contracting Markov chain and the observation process is absolutely regular with geometrically decaying coefficients.
Journal ArticleDOI
Weak convergence of the tail empirical process for dependent sequences
Holger Rootzén,Holger Rootzén +1 more
TL;DR: In this paper, the authors prove weak convergence in D of the tail empirical process for a large class of stationary sequences, where moment restrictions on the amount of clustering of extremes, restrictions on long range dependence (absolute regularity or strong mixing), and convergence of the covariance function.
Journal ArticleDOI
Asymptotic Stability Region of Slotted Aloha
TL;DR: Theoretical evidence and numerical experiments are provided to explain why the proposed approximate stability condition is extremely accurate even for systems with a restricted number of users (even two or three).
Journal ArticleDOI
Asymptotic Theory for Zero Energy Functionals with Nonparametric Regression Applications
Qiying Wang,Peter C.B. Phillips +1 more
TL;DR: In this article, a local limit theorem for the sample mean of a zero energy function of a nonstationary time series involving twin numerical sequences that pass to infinity is given for certain nonparametric kernel density estimation and regression problems where the relevant quantities are functions of both sample size and bandwidth.
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.