  In this paper we consider sample path growth of superpositions of positive
Ornstein--Uhlenbeck type processes (supOU). SupOU are stationary infinitely
divisible processes defined as integrals with respect to a random measure. They
allow marginal distributions and correlations to be modeled independently. Our
results show that the almost sure behavior is primarily governed by the tail of
the marginal distribution. In particular, we obtain a general integral test for
the sample path growth that covers both heavy-tailed and light-tailed
scenarios. We also investigate the tail behavior of the marginal distributions
in connection with the characteristics of the underlying random measure.
