Preface to the Fourth Edition Since the publication of the third edition in 1998, some new developments have occurred. Samuel Karlin died in 2007, leaving a gap at the authorship level and the new designation of authors. In the fourth edition, we have added two new chapters: Chapter 10 on random evo- lution and Chapter 11 on characteristic functions. Random Evolution denotes a set of stochastic models, which describe continuous motion with piecewise linear sample functions. Explicit formulas are available in the simplest cases. In the general case, one has a central limit theorem, which is pursued more generally in Chapter 11, "Char- acteristic Functions and Their Applications." Here the necessary tools from Fourier Analysis are developed and applied when necessary. Many theorems are proved in full detail, while other proofs are sketched--in the spirit of the earlier Chapters 1­9. Complete proofs may be found by consulting the intermediate textbooks listed in the section on further reading. Instructors who have taught from the third edition may be reassured that Chapters 1­9 of the new edition are identical to the corresponding chapters of the new book. We express our thanks to Michael Perlman of the University of Washington and