Spectral decomposition for operator self-similar processes and their generalized domains of attraction

A stochastic process on a finite-dimensional real vector space is operator-self-similar if a linear time change produces a new process whose distributions scale back to those of the original process, where we allow scaling by a family of affine linear operators. We prove a spectral decomposition theorem for these processes, and for processes with these scaling limits. This decomposition reduces the study of these processes to the case where the growth behavior over time is essentially uniform in all radial directions.