This paper, with an extensive bibliography of 89 items, is an advanced survey on parts of probabilistic number theory, describing the important ideas in many papers, for example by Erdős, Kac, Kubilius, Babu, Timofeev, Lyashenko, Philipp, Hall, Tenenbaum, and many others, and, of course, by the author himself.
The paper deals with
– partial sum processes for independent random variables,
– additive functions and functionals on them,
– additive functions and Brownian motion,
– models of other processes with independent increments,
– additive functions on sparse sequences,
– multiplicative functions,
– divisors and stochastic processes.
In the abstract, it is written, that the paper starts “from the invariance principle established by P. Erdős and M. Kac in the fourties and from more general functional limit theorems for partial sum processes for independent random variables.” Then “the development of a parallel theory dealing with those dependent random variables which appear in probabilistic number theory” is described. In particular, the author treats in this survey “results on the weak convergence of processes defined in terms of arithmetical functions”.
For the entire collection see [Zbl 0999.00015].