19. Processes with Independent Increments

Section 19.1 introduces the fundamental concept of infinitely divisible distributions and contains the key theorem on relationship of such processes to processes with independent homogeneous increments. Section 19.2 begins with a definition of the Wiener process based on its finite-dimensional distributions and establishes existence of a continuous modification of the process. It also derives the distribution of the maximum of the Wiener process on a finite interval. The Laws of the Iterated Logarithm for the Wiener process are established in Sect. 19.3. Section 19.4 is devoted to the Poisson processes, while Sect. 19.5 presents a characterisation of the class of processes with independent increments (the Lévy–Khintchin theorem).