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

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

Convex Analysis: (pms-28)

In modern economies, multiple means of payment associated with the exchange of goods coexist. This paper examines one such payment system in an economy with endogenous technological change. It consists of money and a costly accounting system that receives spillovers from new technologies. Positive nominal interest rates are shown to produce welfare losses by inducing a reallocation of human capital into the payment system, and out of the production of final goods and new knowledge. The former substitution produces level effects on output and the latter produces growth effects. At higher levels of inflation, these marginal effects are seen to be weaker. In modern economies, multiple means of payment associated with the exchange of goods coexist. Among the available options that facilitate these transactions are costly accounting systems such as credit and debit cards for households and cash concentration and sweep accounts for firms. These systems ultimately depend upon various methods of implementing electronic funds transfers (EFTs), whose efficiency in turn is dependent upon computer and information systems technology. Therefore, the efficacy of these means of payment in minimising fundamental trading frictions can be enhanced by improvements in the techniques for implementing EFTs, or by the 'spillovers' of advances in computer and information systems technology via their application to the payment system. This technology interacts with monetary policy in determining the structure of the economy's payment system. In particular, as inflation rises, agents respond by selecting a payment system with a mix of media of exchange that is more heavily weighted in favour of costly accounting systems to avoid the inflation taxes imposed on cash transactions.1 This may misallocate resources and thus reduce welfare. Marquis and Reffett (I 992 b) examine one such payment system in a stochastic neoclassical growth model where the costs of operating the accounting system arise from using physical capital to run the system. In this economy, inflation taxes are shown to produce level effects on output even when a cash-in-advance constraint is imposed only on a subset of the consumption goods. This contrasts with Abel's (i 985) result for a deterministic, cash-in-advance economy, where it is shown that the cash-in-advance constraint must also be imposed on investment goods for anticipated inflation * We thank Sheng-Cheng Hu, Peter Ireland, Stacey Schreft, Stephen Williamson, participants at the I993 Winter Meetings of the Econometrics Society, seminar participants at the Board of Governors of the Federal Reserve System, and two anonymous referees for their helpful comments on earlier versions of this paper. The views expressed in this paper are those of the authors and do not necessarily reflect the opinions of the Board of Governors of the Federal Reserve System, or its staff.

New technology spillovers into the payment system

In this paper, we provide a set of sufficient conditions under which recursive competitive equilibrium exist and are unique for a large class of distorted dynamic equilibrium models with capital and elastic labor supply. We develop a monotone map approach to the problem. The class of economies for which we are able to obtain our existence result is apparently considerably larger than those considered in previous work. Additionally unlike previous work, we are able to also prove that this equilibrium is unique. We conclude by applying the new results to some important examples of monetary economies often used in applied work.

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

https://isiarticles.com/bundles/Article/pre/pdf/23306.pdf

Proposes that service quality can be improved by focusing on customer problems. Describes a problem‐centred research program used to establish the types and pervasiveness of customer problems and to evaluate the effect of each on customer satisfaction. Makes it possible to identifycritical problem areas and to establish service priorities accordingly. Offers recommended guidelines for designing and conducting problem‐centred consumer research.

Kevin Reffett

Papers

New technology spillovers into the payment system

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

Focusing on customer problems to improve service quality

A qualitative approach to Markovian equilibrium in infinite horizon economies with capital