The first problem attacked in this paper is answering the question whether all 1/[alpha]-self-similar [alpha]-stable processes with stationary increments are [alpha]-stable motions. The answer is yes for [alpha] = 2, no for 1[less-than-or-equals, slant][alpha]<2 and unknown for 0<[alpha]<1. We single out the log-fractional stable processes for 1<[alpha][less-than-or-equals, slant]2, different from [alpha]-stable motions for [alpha][not equal to]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 [Pi] of slowly varying functions.
stable process stable motion self-similar process with stationary increments fractional stable process log-fractional stable process domain of attraction moving average de Haan's class [Pi]
