  We completely characterize $\Delta$- and local subexponentialities of
positive-half compound Poisson distributions and extend the characterization on
two-sided distributions. Moreover, $\Delta$-subexponentiality of infinitely
divisible distributions is characterized with new conditions, and local
subexponentiality is newly characterized in the two-sided case. In the process
closedness properties of these subexponentialities are derived, particularly
for distributions on $\R$. Most results are obtained by exploiting
monotonic-type assumptions. We apply our results to distributions of supremum
of a random work and a randomly stopped iid sum.
