We have sold 4300 copies worldwide of the first edition (1999). This new edition contains five completely new chapters covering new developments.

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Monte Carlo Statistical Methods

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

Convergence of Probability Measures

We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.

/pdf/general-methods-for-monitoring-convergence-of-iterative-45g63vyekv.pdf

General methods for monitoring convergence of iterative simulations

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1988.tb01721.x

Local computations with probabilities on graphical structures and their application to expert systems

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

Preface SIMPLE MARKOVIAN MODELS: Markov Chains Markov Jump Processes Queueing Theory at the Markovian Level BASIC MATHEMATICAL TOOLS: Basic Renewal Theory Regenerative Processes Further Topics in Renewal Theory and Regenerative Processes Random Walks SPECIAL MODELS AND METHODS: Steady-state Properties of GI/G/1 Explicit Examples in the Theory of Random Walks and Single Server Queues Multi-Dimensional Methods Many-server Queues Conjugate Processes Insurance Risk, Dam and Storage Models Selected Background and Notation.

Applied Probability And Queues

General Methods and Algorithms.- Generating Random Objects.- Output Analysis.- Steady-State Simulation.- Variance-Reduction Methods.- Rare-Event Simulation.- Derivative Estimation.- Stochastic Optimization.- Algorithms for Special Models.- Numerical Integration.- Stochastic Di3erential Equations.- Gaussian Processes.- Levy Processes.- Markov Chain Monte Carlo Methods.- Selected Topics and Extended Examples.- What This Book Is About.- What This Book Is About.

Stochastic Simulation: Algorithms and Analysis

Applied Probability and Queues.

The reserve r ( t ) of an insurance company at time t is assumed to be governed by the stochastic differential equation dr ( t ) = ( μ − a ( t )) d t + σ d w ( t ), where w is standard Brownian motion, μ , σ > 0 constants and a ( t ) the rate of dividend payment at time t (0 acts as absorbing barrier for r ( t )). The function a ( t ) is subject to dynamic allocation and the objective is to find the one which maximizes EJ x ( a (·)), where J x = ∫ 0 ∞ e − ct a ( t ) d t is the total (discounted) pay-out of dividend and x refers to r (0) = x . Two situations are considered: 1. (a) The dividend rate is restricted so that the function a ( t ) varies in [0, a 0 ] for some a 0 a 0 is smaller than some critical value, the optimal strategy is to always pay the maximal dividend rate a 0 . Otherwise, the optimal policy prescribes to pay nothing when the reserve is below some critical level m , and to pay maximal dividend rate a 0 when the reserve is above m . 2. (b) The dividend rate is unrestricted so that a ( t ) is allowed to vary in all of (0, ∞). Then the optimal strategy is of singular control type in the sense that it prescribes to pay out whatever amount exceeds some critical level m , but not pay out dividend when the reserve is below m 
.

Søren Asmussen

Papers

Applied Probability And Queues

Stochastic Simulation: Algorithms and Analysis

Applied Probability and Queues.

Applied Probability and Queues.

Controlled diffusion models for optimal dividend pay-out