Log-fractional stable processes

The first problem attacked in this paper is answering the question whether all 1/-self-similar -stable processes with stationary increments are -stable motions. The answer is yes for = 2, no for 1<2 and unknown for 0<<1. We single out the log-fractional stable processes for 1<2, different from -stable motions for ≠2. They can be regarded as the limit of fractional stable processes as the exponent in the kernel tends to 0. The paper ends with a limit theorem for partial sum processes of moving averages of iid random variables in the domain of attraction of a strictly stable law, with log-fractional stable processes as limits in law. The conditions involve de Haan's class Π of slowly varying functions.