  We establish a new class of functional central limit theorems for partial sum
of certain symmetric stationary infinitely divisible processes with regularly
varying L\'{e}vy measures. The limit process is a new class of symmetric stable
self-similar processes with stationary increments that coincides on a part of
its parameter space with a previously described process. The normalizing
sequence and the limiting process are determined by the ergodic-theoretical
properties of the flow underlying the integral representation of the process.
These properties can be interpreted as determining how long the memory of the
stationary infinitely divisible process is. We also establish functional
convergence, in a strong distributional sense, for conservative pointwise dual
ergodic maps preserving an infinite measure.
