  Superpositions of Ornstein-Uhlenbeck processes allow a flexible dependence
structure, including long range dependence for OU-type processes. Their complex
asymptotic is governed by three effects: the behavior of the L\'evy measure
both at infinity and at zero, and the behavior at zero of the measure governing
the dependence. We establish almost sure rates of growth depending on the
characteristics of the process and prove a Marcinkiewicz--Zygmund type SLLN for
the integrated process.
