Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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 an

Convergence of probability measures

Limit Theorems of Probability Theory.

A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

/pdf/the-strong-law-of-large-numbers-for-extended-negatively-3f4ntams7g.pdf

The strong law of large numbers for extended negatively dependent random variables

Nested case-control (NCC) design is used frequently in epidemiological studies as a cost-effective subcohort sampling strategy to conduct biomarker research. Sampling strategy, on the other hoand, creates challenges for data analysis because of outcome-dependent missingness in biomarker measurements. In this paper, we propose inverse probability weighted (IPW) methods for making inference about the prognostic accuracy of a novel biomarker for predicting future events with data from NCC studies. The consistency and asymptotic normality of these estimators are derived using the empirical process theory and convergence theorems for sequences of weakly dependent random variables. Simulation and analysis using Framingham Offspring Study data suggest that the proposed methods perform well in finite samples.

/pdf/evaluating-prognostic-accuracy-of-biomarkers-in-nested-case-squwn3qvbq.pdf

Evaluating prognostic accuracy of biomarkers in nested case–control studies

Summary

This paper considers a moving average process for a sequence of negatively associated random variables. It discusses the complete convergence of such a moving average process under suitable conditions. These results generalize and complement earlier results on independent random variables. Also, a conjecture for the case of a sequence of independent and identically distributed random variables is resolved and its moment condition weakened.

On the convergence of moving average processes under dependent conditions

A central limit theorem for stationary linear processes generated by linearly positively quadrant-dependent process

Based on the idea of the local polynomial smoother, we construct the Nadaraya–Watson type and local linear estimators of conditional density function for a left-truncation model. Asymptotic normality of the estimators is established under the lifetime observations are assumed to be a sequence of stationary $$\alpha $$
 -mixing random variables. Finite sample behavior of the estimators is investigated via simulations too.

Asymptotic normality of conditional density estimation with left-truncated and dependent data

https://link.springer.com/content/pdf/10.1007%2Fs10959-013-0493-0.pdf

RETRACTED: Convergence of weighted sums for arrays of negatively dependent random variables and its applications

Summary

In this paper, we establish strong laws for weighted sums of negatively associated (NA) random variables which have a higher-order moment condition. Some results of Bai Z.D. & Cheng P.E. (2000)[Marcinkiewicz strong laws for linear statistics. Statist. and Probab. Lett.43, 105–112,] and Sung S.K. (2001)[Strong laws for weighted sums of i.i.d. random variables, Statist. and Probab. Lett.52, 413–419] are sharpened and extended from the independent identically distributed case to the NA setting. Also, one of the results of Li D.L. et al. (1995)[Complete convergence and almost sure convergence of weighted sums of random variables. J. Theoret. Probab.8, 49–76,] is complemented and extended.

Jong-Il Baek

Papers

On the convergence of moving average processes under dependent conditions

A central limit theorem for stationary linear processes generated by linearly positively quadrant-dependent process

Asymptotic normality of conditional density estimation with left-truncated and dependent data

RETRACTED: Convergence of weighted sums for arrays of negatively dependent random variables and its applications

Weighted sums of negatively associated random variables