Functionals on stochastic processes; Uniform convergence of empirical measures; Convergence in distribution in euclidean spaces; Convergence in distribution in metric spaces; The uniform metric on space of cadlag functions; The skorohod metric on D [0, oo); Central limit teorems; Martingales.

This book is devoted to sequential methods of solving a class of problems to which belongs, for example, the problem of finding a maximum point of a function if each measured value of this function contains a random error. Some basic procedures of stochastic approximation are investigated from a single point of view, namely the theory of Markov processes and martingales. Examples are considered of applications of the theorems to some problems of estimation theory, educational theory and control theory, and also to some problems of information transmission in the presence of inverse feedback.

This book chronicles Donald Burkholder's thirty-five year study of martingales and its consequences. Here are some of the highlights. Pioneering work by Burkholder and Donald Austin on the discrete time martingale square function led to Burkholder and Richard Gundy's proof of inequalities comparing the quadratic variations and maximal functions of continuous martingales, inequalities which are now indispensable tools for stochastic analysis. Part of their proof showed how novel distributional inequalities between the maximal function and quadratic variation lead to inequalities for certain integrals of functions of these operators. The argument used in their proof applies widely and is now called the Burkholder-Gundy good lambda method. This uncomplicated and yet extremely elegant technique, which does not involve randomness, has become important in many parts of mathematics. The continuous martingale inequalities were then used by Burkholder, Gundy, and Silverstein to prove the converse of an old and celebrated theorem of Hardy and Littlewood. This paper transformed the theory of Hardy spaces of analytic functions in the unit disc and extended and completed classical results of Marcinkiewicz concerning norms of conjugate functions and Hilbert transforms. While some connections between probability and analytic and harmonic functions had previously been known, this single paper persuaded many analysts to learn probability. These papers together with Burkholder's study of martingale transforms led to major advances in Banach spaces. A simple geometric condition given by Burkholder was shown by Burkholder, Terry McConnell, and Jean Bourgain to characterize those Banach spaces for which the analog of the Hilbert transform retains important properties of the classical Hilbert transform. Techniques involved in Burkholder's usually successful pursuit of best constants in martingale inequalities have become central to extensive recent research into two well- known open problems, one involving the two dimensional Hilbert transform and its connection to quasiconformal mappings and the other a conjecture in the calculus of variations concerning rank-one convex and quasiconvex functions. This book includes reprints of many of Burkholder's papers, together with two commentaries on his work and its continuing impact.

All articles, notes, queries, corrigenda, and obituaries appearing in the following journals during the indicated years are indexed: Annals of mathematical statistics, 1961-1969; Biometrics, 1965-1969#3; Biometrics, 1951-1969; Journal of the American Statistical Association, 1956-1969; Journal of the Royal Statistical Society, Series B, 1954-1969,#2; South African statistical journal, 1967-1969,#2; Technometrics, 1959-1969.--p.iv.

The Handbook of Simulation Optimization presents an overview of the state of the art of simulation optimization, providing a survey of the most well-established approaches for optimizing stochastic simulation models and a sampling of recent research advances in theory and methodology. Leading contributors cover such topics as discrete optimization via simulation, ranking and selection, efficient simulation budget allocation, random search methods, response surface methodology, stochastic gradient estimation, stochastic approximation, sample average approximation, stochastic constraints, variance reduction techniques, model-based stochastic search methods and Markov decision processes. This single volume should serve as a reference for those already in the field and as a means for those new to the field for understanding and applying the main approaches. The intended audience includes researchers, practitioners and graduate students in the business/engineering fields of operations research, management science, operations management and stochastic control, as well as in economics/finance and computer science.

Sequential Methods in Pattern Recognition and Machine Learning

This book covers different topics on optimal design and operations with particular emphasis on chemical engineering applications. A wide range of optimization methods ? deterministic, stochastic, global and hybrid ? are considered.Containing papers presented at the bilateral workshop by British and Lithuanian scientists, the book brings together researchers' contributions from different fields ? chemical engineering including reaction and separation processes, food and biological production, as well as business cycle optimization, bankruptcy, protein analysis and bioinformatics.