Some Limit Theorems for Stationary Processes

I. A. Ibragimov

Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 4, pp. 361-392

Résumé

In this paper stationary stochastic processes in the strong sense $\{x_j\}$ are investigated, which satisfy the condition $$ |\mathbf P(AB)-\mathbf P(A)\mathbf P(B)|\leq\varphi(n)\mathbf P(A),\quad\varphi(n)\downarrow 0, $$ for every $A\in\mathfrak{M}_{-\infty}^0,B\in\mathfrak{M}_n^\infty$, or the “strong mixing condition” $$ \sup_{A\in\mathfrak{M}_{-\infty}^0,B\in\mathfrak{M}_n^\infty}|\mathbf P(AB)-\mathbf P(A)\mathbf P(B)|\alpha(n)\downarrow0, $$ where $\mathfrak{M}_a^b$ is a $\sigma$-algebra generated by the events $$ \{(x_{i_1},x_{i_2},\dots,x_{i_k})\in\mathbf E\},\qquad a \leq i_1\dots\leq b, $$ $\mathbf E$ 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}_{-\infty}^\infty $.