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
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.
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.
Book
Lectures on the Poisson Process
TL;DR: In this article, the authors developed the theory of the Poisson process in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the poisson process.
References
More filters
Journal ArticleDOI
Optimal Solutions to Infinite-Player Stochastic Teams and Mean-Field Teams
Sina Sanjari,Serdar Yüksel +1 more
TL;DR: It is shown that under uniform integrability and uniform convergence conditions, an optimal policy for static teams with countably infinite number of DMs can be established as the limit of sequences of optimal policies forstatic teams with N DMs as $N \to \infty$.
Posted Content
A law of large numbers for limit order books
Ulrich Horst,Michael Paulsen +1 more
TL;DR: A stochastic model of a two-sided limit order book in terms of its key quantities best bid [ask] price and the standing buy [sell] volume density is defined and a limit theorem is proved.
Journal ArticleDOI
Comparison of quenched and annealed invariance principles for random conductance model
TL;DR: In this article, it was shown that there exists an ergodic conductance environment such that the weak (annealed) invariance principle holds for the corresponding continuous time random walk but the quenched invariance does not hold.
Journal ArticleDOI
Convergence of clock process in random environments and aging in Bouchaud's asymmetric trap model on the complete graph
TL;DR: In this article, the celebrated arcsine aging scheme of Ben Arous and Cerný is taken up, using a brand new approach based on point processes and weak convergence techniques, and implemented in a broad class of Markov jump processes in random environments.
Posted Content
Asymptotic analysis of the Forward Search
Bent Nielsen,Søren Johansen +1 more
TL;DR: The Forward Search is an iterative algorithm concerned with detection of outliers and other unsuspected structures in data as discussed by the authors, which has been suggested, analysed and applied for regression models in the monograph Atkinson and Riani (2000).