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
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References
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Journal ArticleDOI
Weak invariance principles for sums of dependent random functions
TL;DR: In this paper, the weak convergence of normalized partial sums of dependent random functions exhibiting a Bernoulli shift structure was proved for functional data analysis, motivated by problems in functional analysis.
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.
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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).