  In this paper, we consider the dynamics of integrable stochastic Hamiltonian
systems. Utilizing the Nagaev-Guivarc'h method, we obtain several generalized
results of the central limit theorem. Making use of this technique and the
Birkhoff ergodic theorem, we prove that the invariant tori persist under
stochastic perturbations. Moreover, they asymptotically follow a Gaussian
distribution, which gives a positive answer to the stability of integrable
stochastic Hamiltonian systems over time. Our results hold true for both
Gaussian and non-Gaussian noises, and their intensities can be not small.
