  In this paper we consider a linear stochastic Volterra equation which has a
stationary solution. We show that when the kernel of the fundamental solution
is regularly varying at infinity with a log-convex tail integral, then the
autocovariance function of the stationary solution is also regularly varying at
infinity and its exact pointwise rate of decay can be determined. Moreover, it
can be shown that this stationary process has either long memory in the sense
that the autocovariance function is not integrable over the reals or is
subexponential. Under certain conditions upon the kernel, even arbitrarily slow
decay rates of the autocovariance function can be achieved. Analogous results
are obtained for the corresponding discrete equation.
