Carleman's classes for stationary processes

The main result of the present paper consists in demonstration of the fact that sample functions of a stationary stochastic process belong, with probability one, to Carleman's class $С\{m_n\}$, if the correlation function of the process belongs to the same class, and if
$$0<D=\inf_y\biggl\{y\colon\varlimsup_{n\to\infty}\frac{m_{2n}}{m^2_ny^{2n}}=0\biggr\}<\infty$$

For processes satisfying the conditions
$$\varliminf_{n\to\infty}\mathbf P\{(\xi^{(n)}(0))^2>\mathbf M(\xi^{(n)}(0))^2\}>0,\quad1<\frac{m_n}{m_{n-1}}<Vn^w,$$
where $V$ and $w$ are positive constants, the converse assertion is proved to be also true.