  We characterize the second order subexponentiality of an infinitely divisible
distribution on the real line under an exponential moment assumption. We
investigate the asymptotic behaviour of the difference between the tails of an
infinitely divisible distribution and its L\'evy measure. Moreover, we study
the second order asymptotic behaviour of the tail of the $t$-th convolution
power of an infinitely divisible distribution. The density version for a
self-decomposable distribution on the real line without an exponential moment
assumption is also given. Finally, the regularly varying case for a
self-decomposable distribution on the half line is discussed.
