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

Convergence of Probability Measures

On robust estimation of the location parameter

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

THE development of molecular genetics, both as a self-contained field and as a body of techniques broadly useful in biologic investigation, has had a profound influence on medical research. The beneficiaries include every discipline in basic science and, at least indirectly, most clinical and applied medical disciplines. Certain technical milestones can be identified over the past several decades that have been particularly important in the progress of the field. One is the discovery of restriction endonucleases, which together with the development of DNA ligation and transformation procedures, led to the ability to clone and thus propagate genes of any organism . . .

https://www.nejm.org/doi/pdf/10.1056/NEJM199001183220307

The polymerase chain reaction. A new method of using molecular genetics for medical diagnosis.

/pdf/pcr-sscp-a-simple-and-sensitive-method-for-detection-of-nbc1c460t2.pdf

PCR-SSCP: a simple and sensitive method for detection of mutations in the genomic DNA.

Soit X n le nombre de comparaisons utilisees par la procedure Quicksort pour trier une liste de nombres distincts Nous demontrons que (X n −E(X n ))/n converge faiblement vers une certaine variable aleatoire Y La distribution de Y est le point fixe d'une contraction et peut etre calculee numeriquement part iteration

/pdf/a-limit-theorem-for-quicksort-20ynffk7l4.pdf

A limit theorem for “quicksort”

In this paper we give an introduction to the analysis of algorithms by the contraction method. By means of this method several interesting classes of recursions can be analyzed as particular cases of our general framework. We introduce the main steps of this technique which is based on contraction properties of the algorithm with respect to suitable probability metrics. Typically the limiting distribution is characterized as a fixed point of a limiting operator on the class of probability distributions. We explain this method in the context of several “divide and conquer” algorithms. In the second part of the paper we introduce a new quite general model for branching dynamical systems and explain that the contraction method can be applied in this model. This model includes many classical examples of random trees and gives a general frame for further applications.

The contraction method for recursive algorithms

A fixed point theorem for distributions

The impact of replication errors on the reliability of polymerase chain reaction (PCR) data is studied theoretically. Practical applications of our results to RFLP analysis and oligonucleotide probing confirm that for practical purposes replication errors can be neglected if a large number of starting templates (e.g. 100,000) is being used. For single locus analysis in single cells, however, the probability of false diagnosis due to such errors is of the order of 1 percent.

Polymerase chain reaction: replication errors and reliability of gene diagnosis.

This paper develops general tools for the analysis of stochastic divide and conquer algorithms. We concentrate on the average performance and the distribution of the running time of the algorithm. As a special example we analyse the average performance and the running time distribution of the (2k + 1)-median version of Quicksort.

Uwe Rösler

Papers

A limit theorem for “quicksort”

The contraction method for recursive algorithms

A fixed point theorem for distributions

Polymerase chain reaction: replication errors and reliability of gene diagnosis.

On the analysis of stochastic divide and conquer algorithms