  Layered stable (multivariate) distributions and processes are defined and
studied. A layered stable process combines stable trends of two different
indices, one of them possibly Gaussian. More precisely, in short time, it is
close to a stable process while, in long time, it approximates another stable
(possibly Gaussian) process. We also investigate the absolute continuity of a
layered stable process with respect to its short time limiting stable process.
A series representation of layered stable processes is derived, giving insights
into both the structure of the sample paths and of the short and long time
behaviors. This series is further used for sample paths simulation.
