  The asymptotic properties of the memory structure of ARCH($\infty$) equations
are investigated. This asymptotic analysis is achieved by expressing the
autocovariance function of ARCH($\infty$) equations as the solution of a linear
Volterra summation equation and analysing the properties of an associated
resolvent equation via the admissibility theory of linear Volterra operators.
It is shown that the autocovariance function decays subexponentially (or
geometrically) if and only if the kernel of the resolvent equation has the same
decay property. It is also shown that upper subexponential bounds on the
autocovariance function result if and only if similar bounds apply to the
kernel.
