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Teoriya Veroyatnostei i ee Primeneniya, 1962, Volume 7, Issue 4, Pages 361–392
(Mi tvp4736)
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This article is cited in 499 scientific papers (total in 500 papers)
Some Limit Theorems for Stationary Processes
I. A. Ibragimov Leningrad
Abstract:
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
$$
\{(x_{i_1},x_{i_2},\dots,x_{i_k})\in\mathbf E\},\qquad a \leq i_1<i_2<\dots<i_k\leq b,
$$
$\mathbf E$ 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 $.
Received: 15.01.1961
Citation:
I. A. Ibragimov, “Some Limit Theorems for Stationary Processes”, Teor. Veroyatnost. i Primenen., 7:4 (1962), 361–392; Theory Probab. Appl., 7:4 (1962), 349–382
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