  In this article, we develop Stein characterization for two-sided tempered
stable distribution. Stein characterizations for normal, gamma, Laplace, and
variance-gamma distributions already known in the literature follow easily. One
can also derive Stein characterizations for more difficult distributions such
as the distribution of product of two normal random variables, a difference
between two gamma random variables. Using the semigroup approach, we obtain
estimates of the solution to Stein equation. Finally, we apply these estimates
to obtain error bounds in the Wasserstein-type distance for tempered stable
approximation in three well-known problems: comparison between two tempered
stable distributions, Laplace approximation of random geometric sums, and six
moment theorem for the symmetric variance-gamma approximation of functionals of
double Wiener-It$\ddot{\text{o}}$ integrals. We also compare our results with
the existing literature.
