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
A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law
TL;DR: In this paper, a universal result in almost sure limit theorem for self-normalized partial sums is established, where S, X1, X2 is a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution.
Journal ArticleDOI
Limit Theorems for Power Variations of Pure-Jump Processes with Application to Activity Estimation
Viktor Todorov,George Tauchen +1 more
TL;DR: In this article, the authors derived the asymptotic behavior of realized power variation of pure-jump Ito semimartingales as the sampling frequency within a fixed interval increases to infinity.
Posted Content
Quasi-stationary distributions for structured birth and death processes with mutations
TL;DR: In this paper, the authors studied the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions and proved that there exists a unique quasi-stationary distribution with maximal exponential decay rate.
Journal ArticleDOI
Optimal control of Piecewise Deterministic Markov Processes: a BSDE representation of the value function
TL;DR: In this article, the authors considered an infinite-horizon discounted optimal control problem for piecewise deterministic Markov processes, where a piecewise open-loop control acts continuously on the jump dynamics and on the deterministic flow.
Journal ArticleDOI
A Functional Combinatorial Central Limit Theorem
Andrew Barbour,Svante Janson +1 more
TL;DR: In this paper, a functional version of the Hoeffding combinatorial central limit theorem is established, and a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the original random process.
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.