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

This ebook is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Language (2nd Ed.). One of the best-selling programming books published in the last fifty years, "K&R" has been called everything from the "bible" to "a landmark in computer science" and it has influenced generations of programmers. Available now for all leading ebook platforms, this concise and beautifully written text is a "must-have" reference for every serious programmers digital library.

As modestly described by the authors in the Preface to the First Edition, this "is not an introductory programming manual; it assumes some familiarity with basic programming concepts like variables, assignment statements, loops, and functions. Nonetheless, a novice programmer should be able to read along and pick up the language, although access to a more knowledgeable colleague will help."

The C programming language

생체의 모델링과 Simulation

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Reversibility and Stochastic Networks

The truncated normal distribution

Many stochastic systems have arrival processes that exhibit clustering behavior. In these systems, arriving entities influence additional arrivals to occur through self-excitation of the arrival pr...

Queues Driven by Hawkes Processes

Understanding how delayed information impacts queueing systems is an important area of research. However, much of the current literature neglects one important feature of many queueing systems, namely non-stationary arrivals. Non-stationary arrivals model the fact that customers tend to access services during certain times of the day and not at a constant rate. In this paper, we analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information with time-varying arrivals. In the first model, customers receive queue length information that is delayed by a constant $$\Delta $$
 . In the second model, customers receive information about the queue length through a moving average of the queue length where the moving average window is $$\Delta $$
 . We analyze the impact of a time-varying arrival rate and show using asymptotic analysis that the time-varying arrival rate does not impact the critical delay unless the frequency of the time-varying arrival rate is twice that of the critical delay. When the frequency of the arrival rate is twice that of the critical delay, then the stability is enlarged by a wedge that is determined by the model parameters. As a result, this problem allows us to combine the theory of nonlinear dynamics, parametric excitation, delays, and time-varying queues together to provide insight into the impact of information on queueing systems.

An analysis of queues with delayed information and time-varying arrival rates

Internet and mobile services often provide waiting time or queue length information to customers. This information allows a customer to determine whether to remain in line or, in the case of multip...

Nonlinear Dynamics in Queueing Theory: Determining the Size of Oscillations in Queues with Delay

In this paper, we provide a new technique for analyzing the nonstationary Erlang loss queueing model with abandonment. Our method uniquely combines the use of the functional Kolmogorov forward equations with the well-known Gram-Charlier series expansion from the statistics literature. Using the Gram-Charlier series expansion, we show that we can estimate salient performance measures of the loss queue such as the mean, variance, skewness, kurtosis, and blocking probability. Lastly, we provide numerical examples to illustrate the effectiveness of our approximations.

Nonstationary loss queues via cumulant moment approximations

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