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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