Some Limit Theorems for Stationary Processes

In this paper stationary stochastic processes in the strong sense are investigated, which satisfy the condition \[ | {{\bf P} ( {AB} ) - {\bf P} ( A ){\bf P} ( B )} | \leqq \varphi ( n ){\bf P}( A ),\quad \varphi ( n ) \downarrow 0, \] for every $A \in \mathfrak{M}_{ - \infty }^0 ,B \in \mathfrak{M}_n^\infty $, or the “strong mixing condition” \[ \mathop {\sup }\limits_{A \in \mathfrak{M}_{ - \infty }^0 ,B \in \mathfrak{M}_n^\infty } | {{\bf P} ( {AB} ) - {\bf P} ( A ){\bf P} ( B )} |\alpha ( n ) \downarrow 0, \] where $\mathfrak{M}_a^b $ is a $\sigma $-algebra generated by the events \[ \{ {( {x_{i_1 } ,x_{i_2 } , \cdots ,x_{i_k } } ) \in {\bf E}} \},\qquad a \leqq i_1 < i_2 < \cdots < i_k \leqq b, \]${\bf E}$ being a k-dimensional Borel set.Some limit theorems for the sums of the type \[ \frac{{x_1 + \cdots + x_n }}{{B_n }} - A_n \quad {\text{or}} \quad \frac{{f_1 + \cdots + f_n }}{{B_n }} - A_n \] are established. Here $f_j = T^j f$, and the random variable f is measurable with respect to $\mathfrak{M}...