  The celebrated Trotter approximation theorem provides a sufficient condition
for the convergence of a sequence of operator semigroups in terms of the
corresponding sequence of infinitesimal generators. There exist a few results
on the rate of convergence in Trotter's theorem under some constraints. In the
present paper, a new rate of convergence in Trotter's theorem in full
generality is given. Moreover, we see that this rate of convergence works well
to obtain quantitative estimates for some limit theorems in probability theory.
