Large deviations of stochastic processes close to the Gaussian ones

Asymptotic expansions for the probability $\displaystyle\mathbf P\{\max_{t\in[0,T]}X_{(n)}(t)>u\}$ when $u\to\infty$ or $u,\,T\to\infty$ are given. It is supposed that the random process $X_{(n)}$ is close to the Gaussian process in some sense and is smooth enough in mean quadratic. As an example of application we consider the central limit theorem for random processes which are smooth in mean quadratic and for the noise-process.