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

Invariant measures under random integral mappings and marginal distributions of fractional L\'evy processes