ABSTRACT
The concept of infinite divisibility was introduced and developed in a very short period and by a handful of people. It started in 1929 with the introduction by de Finetti of infinite divisibility in the context of processes with stationary independent increments (sii-processes); its first development ended in 1934 with the most general form of the canonical representation of infinitely divisible characteristic functions by Le´vy in a paper having the term ‘integrals with independent increments’ in its (French) title. Even more than sii-processes, questions related to the central limit theorem led to the introduction and study of infinitely divisible and, more specifically, self-decomposable and stable distributions. Little, and mostly ‘academic’, attention was paid to the subject during the forties and fifties. Since about 1960 there was a renewed interest in infinitely divisible distributions, stimulated by the occurrence of such distributions in applications, most notably in waiting-time theory and in modelling problems. This led to the development of criteria for infinite divisibility in terms of distribution functions and densities rather than in terms of characteristic functions. This development culminated in the appearance of Bondesson’s book on generalized gamma convolutions in 1992. In the last ten years much attention went to infinite divisibility on abstract spaces; these developments are outside the scope of this book.
