On a criterion of weak dependence

Let $\xi_t$, $t\in Z^1$, be a stationary real-valued random process and let $\mathfrak{M}_a^b$, $-\infty\le a<b\le\infty$, be the $\sigma$-algebra generated by the random variables $\xi_t$, $a\le t\le b$. We say that the process $\xi_t$, $t\in Z^1$, satisfies the $\beta$-mixing condition if for any $A\in\mathfrak{M}_{-\infty}^+$, $B\in\mathbf W\mathfrak{M}_{t+\tau}^\infty$, $\tau>0$, $\tau\in Z^1$,
\begin{equation} |\mathbf P(AB)-\mathbf P(A)\mathbf P(B)|\le\beta(\tau)\mathbf P(A)\mathbf P(B),\qquad\beta(\tau)\to 0,\tau\to\infty. \end{equation}
It is shown that the Gibbs random process under some conditions on the potential satisfies the criterion (1). The main result of the paper is the following
\smallskip Theorem. If the process $\xi_t$, $t\in Z^1$, satisfies the condition (1), $\sigma_n^2=\mathbf D(\xi_0+\xi_1+\dots+\xi_n)\ge C_n$, $0<C<\infty$, and $\mathbf M\xi_0^2<\infty$, then
$$ \lim_{n\to\infty}\mathbf P\left\{\frac{1}{\sigma_n}\sum_{t=m}^{n+m}(\xi_t-\mathbf M\xi_t)<\alpha\right\}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\alpha e^{-t^2/2}\,dt,\qquad m\in Z^1. $$