The Poisson `law of small numbers' is a central principle in modern theories of reliability, insurance, and the statistics of extremes. It also has ramifications in apparently unrelated areas, such as the description of algebraic and combinatorial structures, and the distribution of prime numbers. The law of small numbers is only an approximation. In 1975, a new technique was introduced, the Stein-Chen method, which makes it possible to estimate the accuracy of the approximation in a wide range of situations. This book provides an introduction to the method, and a varied selection of examples of its application, emphasizing the flexibility of the technique when combined with a judicious choice of coupling. It also contains more advanced material, in particular on compound Poisson and Poisson process approximation, where the reader is brought to the boundaries of current knowledge.
Table of contents:
Introduction; Upper bounds; Lower bounds; Random permutations; Random graphs; Occupancy and urn models; Spacings; Exceedances and extremes; Independent and dissociated terms; Poisson process approximation; Appendices; References; Index.
Table of contents:
Introduction; Upper bounds; Lower bounds; Random permutations; Random graphs; Occupancy and urn models; Spacings; Exceedances and extremes; Independent and dissociated terms; Poisson process approximation; Appendices; References; Index.