  It is shown that some convolution semigroups of infinitely divisible measures
are invariant under the random integral mappings $I^{h,r}_{(a,b]}$ defined in
$(\star)$ below. The converse implication is specified for the semigroups of
generalized s-selfdecomposable and selfdecomposable distributions. Some
application are given to the moving average fractional L\'evy process (MAFLP).
