RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1962, Volume 7, Issue 4, Pages 361–392 (Mi tvp4736)

Some Limit Theorems for Stationary Processes

I. A. Ibragimov

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
$$\{(x_{i_1},x_{i_2},…,x_{i_k})\in\mathbf E\},\qquad a \leq i_1<i_2<…<i_k\leq b,$$
$\mathbf E$ being a $k$-dimensional Borel set.
Some limit theorems for the sums of the type
$$\frac{x_1+\cdots+x_n}{B_n}-A_n\quad{or}\quad\frac{f_1+ \cdots+f_n}{B_n }-A_n$$
are established. Here $f_j=T^j f$, and the random variable $f$ is measurable with respect to $\mathfrak{M}_{-\infty}^\infty$.

Full text: PDF file (2597 kB)

English version:
Theory of Probability and its Applications, 1962, 7:4, 349–382

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

Citation in format AMSBIB
\Bibitem{Ibr62} \by I.~A.~Ibragimov \paper Some Limit Theorems for Stationary Processes \jour Teor. Veroyatnost. i Primenen. \yr 1962 \vol 7 \issue 4 \pages 361--392 \mathnet{http://mi.mathnet.ru/tvp4736} \transl \jour Theory Probab. Appl. \yr 1962 \vol 7 \issue 4 \pages 349--382 \crossref{https://doi.org/10.1137/1107036} 

• http://mi.mathnet.ru/eng/tvp4736
• http://mi.mathnet.ru/eng/tvp/v7/i4/p361

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. A. Ibragimov, “Dobrushin's works on Markov processes”, Russian Math. Surveys, 52:2 (1997), 239–243
2. K. Budsaba, P. Chen, A. I. Volodin, “Limiting behaviour of moving average processes based on a sequence of $\rho^-$ mixing and negatively associated random variables”, Lobachevskii J. Math., 26 (2007), 17–25
3. Theory Probab. Appl., 55:3 (2011), 371–394
4. Wintenberger O., “Weak Transport Inequalities and Applications To Exponential and Oracle Inequalities”, Electron. J. Probab., 20 (2015), 114
5. Feng F., Wang D., Wu Q., “An almost sure central limit theorem for self-normalized weighted sums of the ? mixing random variables”, J. Math. Inequal., 10:1 (2016), 233–245
6. Wintenberger O., “Exponential inequalities for unbounded functions of geometrically ergodic Markov chains: applications to quantitative error bounds for regenerative Metropolis algorithms”, Statistics, 51:1 (2017), 222–234
7. Rio E., “Asymptotic Theory of Weakly Dependent Random Processes”, Asymptotic Theory of Weakly Dependent Random Processes, Probability Theory and Stochastic Modelling, 80, Springer-Verlag Berlin, 2017, 1–204
8. Kanaya Sh., “Convergence Rates of Sums of Alpha-Mixing Triangular Arrays: With An Application to Nonparametric Drift Function Estimation of Continuous-Time Processes”, Economet. Theory, 33:5 (2017), 1121–1153
9. Theory Probab. Appl., 63:3 (2019), 479–499
10. Ibragimov I.A. Lifshits M.A. Nazarov A.I. Zaporozhets D.N., “On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables”, Vestn. St Petersb. Univ.-Math., 51:3 (2018), 213–236